I'm not too experienced a player (maybe 500 hrs?) but I find I play progress better than the other two trees. I play on emperor.
Early:
For policies, start with the right side and go for Equality. The gold generation allows you to support tons of cities before markets. I often have 5 cities while my opponents have 2-3 (any more and happiness can become an issue). My army is usually the smallest though. Use gold to buy units to deter opportunistic civs. Once markets come up you'll be able to purchase a more respectable army if needed.
For techs, grab Pottery immediately and focus on expanding. If you have some kind of gold generation advantage you can delay Trade for other techs. Other techs can help you select better expansion locations. You'll want markets before your buildings/units bring you into the negatives though, so plan ahead. Iron working is also important for the forge, which will help your cities build things and later grab your a wonder with the Great Engineer.
I like to build shrines, because they are maintenance-free and religion is a great boon to expanders. Monuments are important to keep up in policies though, so I usually build shine first in most cities, but monument first in others.
You generally want to be building buildings rather than an army because you don't want unhappiness problems. Because of this nearby Authority civs will have a natural advantage over you -- consider trading them a luxury resource for free for the diplomatic benefit. Tradition civs are safer to expand towards, but they can still have an army advantage if they didn't spend hammers expanding.
As progress you have the hardest time scoring an ancient/classic wonder because of how many settlers you built. I would not attempt it unless you have some kind of production advantage (luxury resource mines, or being Egypt or something). Your policies will fall behind a little as well, further hurting your chances at wonders. It's not too bad though as you should be able to nab one with your first Great Engineer. By medieval you will have good culture generation from building buildings and maybe even a city with nothing to do. You should be able to competitively build wonders now, as other civs will have the extra cost from existing wonders. If you are going to build a wonder, I would recommend Oracle. The free policy can snowball you into a wonder building civ, and I find AI's are usually competing for Angkor Wat and Colossus at this time.
Mid:
Expand slowly, but don't feel pressured to if your happiness/military can't handle it. Around this time I find other civs catch up or even surpass me in city count (5-6). Just stay calm and focus on building high quality cities and expanding carefully. You can dump your gold into an army or start-up cities, depending on the situation. Go ahead and build wonders with your capital or cities with a good amount of buildings. Knights are a terrific unit that you can probably utilize better than the AI's. You can brute-force civs still stuck in classical, or you can stall and confuse medieval civs.
It's small, but this is where Progress starts snowballing. Your focus should be securing small advantages while making sure your civ doesn't collapse on itself. I don't like to go full-conquest at this time, because it delays the snowball, but a progress civ can definitely start warmongering at this stage. If you choose to do so, the best time is before enemy civs unlock Tercios, which negates the usefulness of Knights.
If you have been building your buildings, you should have the infrastructure to expand faster around the renaissance. If you teched to pioneer (Banking), you're in a better position to take advantage of this. Look into Astronomy as you are probably running out of land and looking to expand overseas soon. Economics is a tech you want to build buildings even faster, but Navigation is probably the better choice. With your decent city count, you can start building your navy, which will help you protect your overseas cities, and gain an absolute advantage on any civ with no navy. I usually have 8~ cities at this time, and looking to expand a lot more.
In Industrial you should be near the top in terms of city count, maybe second or third to a runaway civ. This means you can have a large standing army that can give Authority civs pause. From this point on Progress feels pretty terrific. Instant yields from cities building buildings every turn, large standing army, good happiness, gold, etc... While other ancient era policies start to become lackluster, you'll always have that amazing 20% production for building buildings and tons of scaling bonus yields. While Authority has to think about happiness and infrastructure, and Tradition worries about their long-term military, as Progress, you are free to focus on any aspect of the game that pleases you most.
The only thing to stay aware of is sudden swings in happiness. You can lose 20~ happiness in a single turn due to how many cities you have. Always pay attention to your cities' needs.
Late:
Don't really have a guide to play this cause usually I just do whatever is best for my civ at this point. The sheer amount of stuff that having a wide progress civ nets you is enough to support just about any playstyle.
tl;dr
- authority and tradition will outperform you early, play nice
- expand while maintaining happiness and gold
- build buildings
- your mid-late game is superior
from forests make wonderful progress 
Try it.
