It's not really possible to give you any concrete information, since we don't know how the future will turn out, but here's my thought process. Basically, I'm going to focus on the FTL stuff, since everything else is kind of cake next to that.
TL;DR. I'd estimate 10-50 years for NASA to find a working FTL prototype at tiny scales, 50-100 years to get enough funding and energy to test full-scale prototypes, 100-200 years to get colonies around the solar system, 100-200 years to put people in another star system, and 100-300 years to have a real extra-solar civilization. Total timeline is then 360-850 years if Earth learns to work as a team. Timeline could be thousands of years to never if we don't.
Energy Requirements for Proposed FTL
FTL may never be possible. All of our current models say it's either impossible or at least not practically meaningful (not much point in traveling faster than light if what shows up is a nuclear explosion instead of a person).
So let's assume NASA's current research on warp drives pans out. Last I heard, just opening a warp bubble was something like a bus worth of antimatter. Wikipedia says a school bus is 4500 to 16000 kg. From $e=mc^2$, we can see that's $4\cdot 10^{20}$ to $1.4\cdot 10^{21}J$ 1. Let's split the difference and call it $10^{21}$J = 1 zettajoule. WolframAlpha tells us that's about as much energy as we get in 1.6 hours of sunlight across the entire surface of the planet, or $1\over 7$ of the energy in the world's oil reserves or $1\over 6$ of the energy in the world's natural gas reserves. Effectively, we'd need every last scrap of natural energy on the planet just to test the thing. Our only options will be magic or solar power.
Current solar panels can operate around 40-45% under ideal circumstances. Let's say we push that up to 50% for mass-produced panels somehow. Now, near Earth, we have about $1350{W\over m^2}$ $={J\over s\cdot m^2}$. So we can get a lot of energy by either waiting a long time, or building a really big solar farm, or both. Let's say 1 year is an acceptable timeframe to charge a warp battery. That's $50\%\cdot 1350{J\over s\cdot m^2}\cdot 86400{s\over day}\cdot{365.25(ish) days\over yr}\cdot 1yr$ $=2.1\cdot 10^{10}{J\over m^2}$. Ok, so we need $10^{21}J\over 2.1\cdot 10^{10}{J\over m^2}$ $=4.8\cdot 10^{10}m^2$ of solar panels that are constantly pointing straight at the sun. Or around 76% of West Virginia ($6.3\cdot 10^{10}m^2$).
Now, it seems likely we would want all those solar panels in orbit around the Sun, so they're getting maximum insolation at all times. This blog says solar panels are around $2$ to $3{lb\over ft^2}$ $=12{kg\over m^2}$. So we need to lift $12{kg\over m^2}\cdot 4.8\cdot 10^{10}m^2$ $=5.8\cdot 10^{11}kg$ of solar panels into orbit. According to this SE answer the cheapest we can do is about $\$2000\over kg$ into orbit. That's $5.8\cdot 10^{11}kg\cdot {$2000\over kg}$ $=\$1.2\cdot 10^{15}$ or about 100 times as much as U.S. citizens make in a year (before paying taxes, bills, etc.).
Then we need to take solar panel production cost into account. This PV site says the panels themselves are 30 to 35% of the installation cost, and this other PV site says a $28 m^2$ system costs around £7000 = \$10000, or about $\$340\over m^2$. This is insignificant compared to the $\$24000\over m^2$ we need to get it into orbit, so there's no need to add them. However, it does suggest that early FTL might be better powered using ground-based solar power that takes longer to get the same energy but costs about 1% as much to generate.
Logistics of Getting Energy
Now, I'm not an economist, but those are some pretty hefty numbers. There's no way the world just says "sure, we'll all pitch in 100 times what the U.S. makes in a year". I'm not even sure we have that kind of resources to throw around. Realistically, we're looking at maybe 1% of that being put towards ever-bigger solar fields over many decades, but multiple countries may be helping out. So let's say 2% to be optimistic. That's 50 years of pumping money into a solar panel farm just to get a maiden voyage going.
But there's a problem. Solar panels aren't immortal, so we have to take into account replacement costs. Unfortunately, that math looks really complicated so I'm going to skip it. It probably doesn't matter too much since most panels are rated at 80% after 25 years (says these guys), but it will take longer to reach our goal. On the other hand, solar technology will get cheaper over time. So from the first time NASA is able to demonstrate very small warp bubbles are possible, we're probably looking at 50+ years to get the first human-sized spaceship going on a test flight. Given the nature of politics, it could take a lot longer than that.
Expansion and Colonization
From here, we need to expand. The problem is, there's really not much reason to explore space except to say we did it. It's really, really expensive, with very little payback. So early FTL is going to be "Yay, we proved FTL is possible!!!!!" followed by a very long period of practically nothing. Eventually, rich people will start paving the way for weddings on Mars, basketball tournaments on Europa, etc. As mentioned in articles like this one from the UK space people, we do get some return on investments, in that we're producing jobs by hiring engineers, cooks, janitors, you name it. But we can't just throw quadrillions of dollars at a project and expect it to instantly give a return. It will take a long time to get anything going in earnest.
That said, warp bubbles aren't inherently FTL. They can presumably be used for sublight travel as well, and we may find ways to do it for cheaper. If we can turn the 2.5 year round-trip to Mars into a 2.5 week trip (0.07% c), a lot more people are going to be willing to head out there to help with whatever science stuff we want. And if we can find sources of natural resources in asteroids or other planets, we may be able to get some return of investment that way (this random article suggests asteroids may be highly lucrative for rare-Earth metals -- it also has some ideas about space-based solar power).
Still, given how slowly the space programs have advanced so far, we're probably looking at 100+ years before we really get anywhere crazy. I would expect it to take a few centuries before we really have civilizations beyond Earth, although that's really just a guess.
FTL Communication
If we can do FTL with ships, it stands to reason we can find ways to do FTL with communications, though the methods and speeds will be greatly dependent on the type of FTL. A portable wormhole could allow a tiny beam of light to transmit information nearly instantly once the connection is established, while some kind of warp bubble would operate more like the current postal service. Still, I would expect FTL communications to evolve more or less concurrently with FTL travel.
Beyond Sol
Up until now, I've been assuming the energy requirements for FTL were about the same as just creating the warp bubble (I really don't know so it could make a huge difference). But trying to cross interstellar space is going to require enormous amounts of energy. Space. Is. Huge. Pluto is about 40 AU from the Sun. Proxima Centauri (the closest star to us) is about 270000 AU away, or 6640 times the distance. More realistically, we'll be heading to Jupiter, at only 5.2 AU, making P Centauri 51000 times as far. Not only do we need more energy to get there, but we need to do it faster. Not many people are going to be up to the task of spending 40 years getting there at 10% c. So if warp drives are less efficient at higher speeds, it makes the energy requirement exponentially worse.
So one trip to the nearest star is going to take an enormous amount of effort compared to sending people back and forth to the gas giants a few times a year. If we get FTL going on a regular basis in 300 years, it will probably be another 100+ years before we make any real progress on extra-solar expansion, and a century or two more before we have actual colonies founded there.
And honestly, I'm probably being pretty generous. It's quite possible it takes a couple thousand years for this timeline to happen. But that's the problem with trying to predict the march of technology, especially regarding technologies that may not ever be physically realizable.
1Technically, we'd need twice that, since there's antimatter and an equal amount of normal matter getting annihilated, but I already made it through a bunch of calculations and it's all roughly estimated anyways, so I'm not going to bother fixing it.
