The "free" stuff they build isn't actually free if you don't tax the rich. As long as you don't outright build factories/railways yourself and you only use the invest function to build factories/railroads, you get the same discount that capitalists do when constructing things. When buildings railroads for instance I still tax the rich at 100% but I use the money I get from them to invest in their railroad projects instead of building my own on the map. You end up getting literally the same thing for the same price that way, so it makes no difference whether you let your capis build it or you take their money and build it for them. When it comes to factory building though, I will spend the extra money to build my own instead of investing in AI projects because the AI always picks the most unprofitable factories.
I think you're totally right. Depending a bit on which country your playing, but all in all you long term goals should be:
1) strong research
2) high population to support your armies with 5% soldier pops (not just through expansion)
3) strong economy to support your army and industry score
Unless you start a game as the UK or another high literacy country there's a couple of things you need to fix first. You lower spending on the military, land and naval stockpiles and the administration. Then you'll max out education spending and max out taxes and tariffs to get your economy in the green again. If that's enough, you can then go to promote clergy up to 2% and then up to 4% in as many states as possible (if there's a serious population disbalance than focus on getting clergy up to 4% in your most populous states first). If you're still in the red, you will have to promote bureaucrats first though and probably switch from time to time between bureaucrats and clergy. If you got all your clergy up to 4% you'll probably switch to bureaucrats unless it takes so much time to get your clergy up that your bureaucrats went up by themselves.
For research, you first take Ideological Thought, Idealism, Medicine, culture techs and then you're finally allowed to take Freedom of Trade if you didn't start with that tech. Only exceptions is that you really need freedom of trade to keep your clergy financed, then you're allowed to take it earlier.
Taking Freedom of Trade will usually fix your economy so that you can start saving money for your first factories, usually cement, glass, liquor or wine. Don't lower your taxes, keep all on max. If you have the ability to switch parties, keep a party in power that supports the military, lets you subsidize factories, but has otherwise as little involvement in the economy as possible. When you have money to build factories, switch parties to one that lets you invest in the economy directly, then switch back.
Manpower is usually out of balance in the beginning and will go down a lot. If you have irregulars (and are able to build regulars) destroy all of them that are not red and build as many infantry as possible. If you don't have enough money to construct new units, that is okay, just lower the third stockpile slider till you can afford it. This way when your manpower goes down and you're not able to support your units anymore, you will still have them and are able to use them in a war (or pretend to be stronger than you actually are).
Once you've taken these important first steps try to find an ally (such as the UK) to invade weak targets (such as Tunisia or Johore) for you. If that's not possible, attack yourself. The sooner you start justifying your first CB, the quicker you start burning infamy (burning as much infamy as possible during the game is very important).
Once you've fixed your economy, clergy levels, researched important techs, built your first factories, reformed healthcare you'll see that your situation improves. Promote craftsmen and clergy for a while, then get your soldiers to 5% in all states and get a tech lead in military tech. If you have money to spare, first increase your military stockpile so that you can have all your units ready to fight at any time, then increase military spending to increase manpower and maybe increase administrative spending to around 11% or 21%.
Next goal will be to lower your tariffs. Tariffs hurt your economy by making goods for your factories more expensive. Keeping high taxes helps you do healthcare reforms. Once you got your healthcare and education reforms through and you still have money left, lower the taxes on the poor since they will be the most affected by taxes. Never ever lower taxes on the rich. They will pay a lot of taxes and will still be able to start any construction instantly.