Assuming Rome never fell and the empire was never divided, how far could it have expanded? What would it take for them to have an empire whose size is similar to that of the british or Spanish empires? What advancements in science and technology would have to be made in order for this expansion to happen and how would they fix the communication problems?
By the 1st century CE, when emperor Octavian put the kibosh on further expansion, the Roman Empire had reached its "natural" boundaries.
- The the south it was bounded by the Sahara.
- To the west it was bounded by the Atlantic.
- To the north it was bounded by the Rhine and the Danube, beyond which there were only fierce (and dirt-poor) barbarian tribes and land unsuitable for the kind of agriculture practiced by the Romans. Emperor Trajan expanded the empire into Dacia in the 2nd century, but only because he basically had to.
- To the east it was bounded by the Persian empire, easily as large, as rich and as powerful as the Roman empire; the war between the Romans and the Persians lasted for some seven or eight centuries, until both were defeated by the Arabs. Note that neither the Romans nor the Persians aimed to conquer the other empire; all the Romans wanted was Mesopotamia, and all the Persian wanted was the Roman Levant.
Furthermore, the question is profoundly misleading; "assuming [...] the empire was never divided" is an astonishing assumption. Assuming the empire was never divided is tantamount to assuming that the Roman leadership was stupid enough to condemn the rich and functional eastern part of the empire to partake in the chaos and dissolution into which the poor and disfunctional western part was sinking. The empire was not divided for fun: it was divided because, when it became clear that the western part was unsalvageable, the empire had to be divided if the eastern part was to survive as a functional state.
So, what is to be done?
You have two centuries, the 1st and the 2nd century of the common era, to effect fundamental changes in the course of history; by the 3rd century it will be too late.
You must formalize the position of the emperor.
Remember that most of the Very Important People whom we call emperors, Octavian, Claudius, Vespasian, Titus, Trajan, Hadrian, Marcus Aurelius, were not officially emperors. In their time, the empire was officially a republic, and those people ruled it by accumulating key positions; they were speakers of the Senate, and great tribunes of the people, and great priests, and commanders in chief of the army.
You must introduce an official mechanism for succession and for the transfer of power.
Since the empire was officially a republic, there was no mechanism for the transfer of power from an emperor to the next, and for the selection of the next emperor. Sometimes things went smoothly, for example during the glorious 2nd century, when five good emperors formed the strangest dynasty in history (they were not biologically related, but rather each of them adopted his successor); most of the times the transition from an emperor to the next was a messy affair.
You must do something to lengthen the life of good emperors (for example, Titus) and avoid stupid or plain unlucky emperors (for example, Domitian and Commodus). (I won't mention disasters such as Elagabalus, because as I said the 3rd century is already way too late.)
You must do something bold and decisive to stop the hemorrhaging of silver to the east.
The Roman empire traded with distant India, and with even more distant China, bringing in goods such as silk and cotton and spices and exporting mostly silver. Now, the Roman empire was not especially rich in silver, and this constant hemorrhage did nothing good to the finances of the empire.
You must do something to fix the massive imbalances in the economy of the empire.
The empire was a very diverse place, and especially the north-western parts of it had severe economic problems. Something must be done to prop up the western part of the empire, and most definitely something must be done to accelerate and facilitate economic activity.
You must arrange a tragic childhood accident for Paul of Tarsus.
The beautiful but intollerant new religion introduced by Paul on the basis of an obscure Jewish sect was a stress and a distraction which the empire could very well do without.
What is the goal?
The goal is to keep the empire functional and avoid the Crisis of the Third Century. If you avoid it, and manage to get into the 4th century with a functional Roman state, then you will be well placed to weather the Migration Period, and possibly find a way to integrate the migratory Germans into the empire, possibly even expanding the empire to the north into Germania.
Then, if you somehow handle the barbarian migration, you will have a functional empire by the 6th century, when the plague will come. If by that time you have successfully introduced measures of public sanitation and quarantine, and you keep the empire puttering along, then you can hope to have it develop the economic, social, political and technological features required to launch the Age of Discovery ahead of time and propel it into a world empire.
Not an easy task, but, you asked.
I've been five minutes thinking about answering your question or flagging it as primarily opinion-based. There are too many details to fully answer your question, but I can give you two points to think about.
The Roman empire stopped expanded voluntarily. It is not that they couldn't conquer more, is that they didn't want to. Why? Because it was not profitable. The Roman Republic conquered all of Italy against cultures as well developed as themselves, then absorved the Carthaginian empire, then the rest of the Helenistic empire that haven't been took by the Parthian empire. From then on, the rest of the world was a savage jungle without taming, inhabited by hostile tribes. The iberian peninsula was conquered because of the second punic war, the Gaul because of Ceasar's political ambitions, Britain for Claudius' whim. They lost a ton of money and resources turning those uncivilized territories in prosper Roman provinces. Taking a jungle and turning it into orchards and crops, making roads through forests and marshes, building cities... it all costs money, much more money than it can be extracted from the territory, even if the land is paved in gold. It took centuries before those provinces were prosper enough to make a profit. Spain took tonnes of gold and silver from America and ended poorer and less populated than it was in 1492. That's why the British empire didn't try to build up an empire "à la" Roman or Spanish style: instead they expended as little money as possible, building only a road from the gold mine to the harbour. The Roman empire couldn't gain any money from conquest, except against the Parthian empire, which takes us to the second point.
There wasn't enough technology gap between Rome and its enemies. The conquistadors benefitted from steel and horses - and virii, involuntarily; though disease didn't help with the conquest, it quenched any spirit of rebellion the natives could have had - while the British empire had modern fire weapons. Roman armies were undefeatable in open battle and they could take any fortress, but they needed big numbers to do so. They couldn't conquer vast territories with a few men like Spaniards or British did, which reinforces the first point: it was too expensive to conquer, for little gain. Cost-wise, their only option was Parthia, but Parthia was a tough nut to crack, its battle tactics and terrain being very unfavorable to Roman legions.
Now, if you want the Roman empire to grow bigger than it was the only sensible option is going the Alexander III way: eastward. Despite all the hype with the macedonian phalanx, it was heavy cavalry which granted Alexander the Great his victories throughout the world. He and his Companions pierced through the enemy lines, disecting their armies into little pieces, then turned around to smash them against the pikes of the phalanx. The problem is, it is the Parthians who had learnt the lesson, not the Romans. If Rome had to prevail in this fight, they needed to develop a much bigger and more powerful cavalry, including heavily armoured cavalry and cavalry bowmen.
As you state, communication and transportation are the main limiting factors in keeping an empire vital and united.
Though recent discoveries confirm that the logistic of the Roman empire was quite effective (French oak boards have recently been found in a roman house in Rome), they were probably stretched to their limits when the empire reached its maximum size.
Spanish and British empires could rely on much better navigations techniques capable of crossing the oceans, while the Romans were still stuck with what was basically coastal navigation.
They also had more advanced weapons than the people of the countries they were conquering, while the Romans had only to rely on better tactics and strategies, but were using swords and bows like their counterparts. When you are using cannons and guns against spears and bows you can better use the small numbers you necessarily have.
Personally i think logistics would be the main problem. If you look at Britain it eventually started crumbling because it was spread out to thin.
It would become hard to to enforce an empire on colonies when you won't hear of them rebelling quite some time after the rebellion already started. Then the issue of moving your troops to the rebelling location fast enough to stomp it out before it spreads. With the downside of moving your enforcers creating new weak spots in your empire.
To fix this Radio would be the first real option if you plan on taking whole of Eurasia and Africa but that was still a long way down the road. Any other option would be splitting it up with the use of puppet governments who are able to handle their local problems themselves.
The Roman Empire (and British and Spanish Empires) took territory and ruled by conquest. If you're going to follow this path then you need to keep troops all over the place to suppress rebellion and maintain the peace. These troops need to be supplied, they need to be refreshed, they also need to be from somewhere else in the empire* so there's less risk of them changing sides in a rebellion.
All of this requires logistics.
However if you can get people to consent to being part of the empire, actively opt in to your way of doing things and the wealth or privileges it brings you can have a much larger empire, with local troops, and your logistics and communication needs are much reduced.
You'd also need improvements in navigation, the reason the British and Spanish empires could spread so far is that they had access to the Americas, Southern Africa, Australia etc. Places the Romans either couldn't get to or would struggle to get to consistently.
*standard Roman procedure
I'm no historian, but I'll give it a try.
One of more obvious reasons why Rome fell, its overflow of uncivilized tribals, germans especially. It's hard to keep your mostly civilized state when most of your citizens are just barbaric brutes. Fortunate Rome had its ways to deal with problems like that, look at the Gauls, they've started as conquered barbarians, but thought course of ages they've had become more civilized and and hellenized. It's probable that same would happen to germans, and it happened really, but not in big enough scale, there was simply way to much of them in to short period of time. If you want to change that, just erase Attila, and then german tribes have no really big reason why attack the Empire all at once. Give it some time, and some ambitious general would conquer their land and later start integrating them.
Now when we have all those germans under our yoke, we have shit tons of very war liking individuals. What we do with guys like that? We send them to some war, best to conquer someone. What would be next target? Probably finishing Albion (aka British Isles), later we don't have much to conquer in Europe, only things left are central Europe, with sparsely concreted slavs tribes, and skandinavia with snow, so nothing to search there. Same thing with Africa everything that was conquerable and wasn't sand was already conquered. So next, and only direction is Persia, with those pesky Sassanians, waging war there would be undeniable hard, but we have spirit of Great Alexander at our side, and those shitloads of angry german guys. After beating Persia, there would left no external dangers for Romans, save for some horsemen, but i propose to deal with them in byzantine way, pay some of them to fight rest.
Nice, we've got everything that was to be got, what now? Now we need to keep everything in check, the worst part. Fortunately Rome had some tries to that, splinting their empire and giving rule to several emperors. So just add extra sub-empire in persian land with somewhat Persian ruler, maybe some royal marriage or something? But most important thing we would have to do is to continue enforcing hellenization everywhere, like universalism later, and hope those weird fish guys from Judea stop spreading their dangerous ideas. Who in heavens would wont to have pacifist religion when you have such military?!
P.S. Sorry for not so good English