Ultimately, money is used to buy people's time.
You might think you are paying for a road; in fact you are paying for people to spend time with shovels and jackhammers, people to quarry stone and drill for oil, people to build and maintain the machinery, and people to transport the raw materials to the site. Not to mention yet more people to decide precisely where and how it should be built in the first place.
Some people value their time more than others; there are only so many waking hours in each day, and fatigue sets in quickly if you try to stretch them, or if you try to work through too many days in a row. Most people will try to earn as much money for their time as their education and training permits.
You can influence people to spend their time on projects that are important to you, by offering contracts to do so at a high enough price. Some (possibly much) of that money will be spent on supplies; the rest will be distributed among the workers and managers on the spot. At this stage, where the money comes from is irrelevant; if you control the Treasury, you can print it out of thin air, and it will be accepted at face value.
The above marks about as far as your current understanding reaches.
One must also understand that people working on your project are not working on other projects as a consequence. Other roads will develop more potholes, farms will go uncultivated, cars and computers will not be manufactured - because the people who would normally see to them are instead working on this new project of yours that earns them more money.
Yes, unemployment has also gone down a bit, but not everyone is inherently suited to construction work, and there's a finite supply of the unemployed. In fact, in many places the construction industry is concerned about a shortage of skilled labour. The same goes for the mining, chemical, transport and high-tech industries which feed into your project.
For a single project that's not a very large effect, but it adds up when you initiate a global programme of them. The result is that the supply of goods goes down a bit.
All these newly-enriched people then want to spend their money in order to benefit from it. To begin with, they buy food - construction work is hungry work, after all. With financial security, they move to larger homes, start families, and buy cars and computers and kitchen appliances. They travel to see the world, and need good roads to do so.
In short, demand for goods goes up a bit. But supply of goods has been going down at the same time. The net effect is an increase in prices and a reduction in consumer satisfaction.
The increased prices help to rebalance the economy. The price of food goes up, so being a farmer becomes worthwhile again. Profit margins on gaming computers go up, so chip fabs shift production to CPUs and graphics cards instead of industrial microcontrollers and sensors. Car manufacturers increase their wages and salaries, to bring in enough labour to increase production to meet demand.
Workers begin to leave your project behind, because they feel you're not paying them enough any more - even though your terms are still nominally as generous as they originally were. In order to entice them back, you must increase your own offers still further - and print yet more money to fund them.
This is a vicious cycle. This is how inflation happens.
If it happens quickly enough, the price increases become so noticeable that the perceived value of your currency goes down even faster - people start charging higher prices because they believe prices will increase further before they can spend their earnings - which leads directly to hyperinflation.
The root cause, though, is that you've been increasing the money supply faster than the supply of work hours and productivity. To avoid it, you must fund your projects without increasing the money supply.
Taxation is one of the traditional answers to this problem. Conveniently, one of the major expenditures normally funded by taxation - the military - can now be downsized, since you are so confident that wars will no longer occur (though I suspect you might be wrong in practice). So you can keep taxation roughly constant, and redirect trillions of dollars per annum from the military-industrial complex into public works.