There are many ways to win GtR, but several basic ideas will provide a solid foundation to your game:

1) Get a building completed quickly
The initial two influence points everybody starts with can quickly become a difficult constraint, so it is important to raise that limit as soon as possible by completing a building. The influence boost is the primary concern here, so the specific building you first attempt to complete is not really relevant. Yellow buildings are the easiest to finish. Brown buildings also require one material, but are a little slower because you also need those brown cards (or a Jack) to lead or follow a craftman role. Red and grey buildings cost two, but the reds are also slightly easier, because the Architect role is grey. Avoid starting a blue or purple building as the first one unless your initial hand contains three of that color or contains the Villa.

2) Don't hoard cards
The player that has the most cards flow through their hand has the best chance of finding the buildings that work well together and finding the right materials to finish those buildings. If you can't use a card now or soon, find a way to get rid of it. The cheap Latrine is excellent for this task, as is the Circus. Other good buildings include the Bar and Atrium, allowing you to play otherwise useless cards directly from your hand. On a similar note, try to empty your hand as much as possible before refilling, but balance that with the possibility of missing key actions. Accurately predicting your opponents leads is important here.

3) Merchant, Merchant, Merchant (when your opponents can't)
Getting resources into your vault is the most complex task in GtR and the most lucrative. Vault VPs and Bonus VPs generally account for two-thirds of the winner's score, so don't fall too far behind in the merchant race and always keep an eye out for opportunities to lead a solo or near solo merchant. Some examples:

-Your opponents' stockpiles are empty.
-Your opponents' vaults are full.
-Your opponents' don't have Jacks or Merchants to follow you, such as immediately after a Legionary role.
-You can feed your vault during a different role, i.e.
--Legionary, with the Colosseum
--Patron, with the Bath
--Craftsman or Architect, with the Bath and Garden

Also, look for opportunities to take more merchant actions than your opponent and deny them the same capability. Primarily, this centers around acquiring the elusive Merchant client, so keep them out of the pool if you can't quickly recover them. If you are late to act, avoid leading or following a Merchant with a blue card; use a Jack or the Circus instead. If you are leader next, and have the means, following with a blue to then hire the client can be a strong move. Other methods of playing multiple merchants include building a Palace, the Palace-Circus or Palace-School combos, or the imperial combo of Palace-Circus Maximus.

4) Avoid running your stockpile completely dry
Having materials in your stockpile gives you options when opponents lead an Architect or a Merchant, so don't give them the chance for a free play. Use the Laborer and Legionary roles to fill your stockpile when others can't. A nearly dry pool is begging for an opportunistic Laborer lead, but be aware of opposing Docks sniping your action. A Tower helps here, and a timely Storeroom will buffer your stockpile for the rest of the game.

5) Take clients for roles your opponents are likely to lead
You can often accomplish more by following your opponents than by leading a role yourself. This is because having the right clients gives you the ability to both Think and then take action during a single turn. For example, if the initial pool is one brown and three greys, it may seem tempting to grab the only Craftsman client, but then you'll just watch your opponents completing buildings with Architects instead. Having a large number of one client type is great when it's your turn, but your opponents will almost never lead that role to you.

6) K.I.S.S.
That Gate-Archway-Stairway-Scriptorum-Sewer combination with a host of Architects certainly looks neat as you are building it, but the game will almost always end just a moment too soon. The game is full of insanely powerful multi-building combos that set you up for huge late-game plays, but the more buildings that you start, the easier it is for your opponents to finish the game by picking up the rest of the Sites before you've scored any real points. Simply finishing many buildings is not a very efficient method of scoring points...you'll need some time to use those buildings' abilities and take advantage of the extra influence they grant.

7) Manage your Jacks
Having a Jack will save your game when you need them, but holding and recovering Jacks eat up a large amount of handspace and time. Unless you have a solid hunch the next player will lead a role you must follow, refrain from taking a Jack before refilling your hand. On the other hand, if you must lead a specific role on your turn, make sure you will have a chance to take a Jack. If the players preceding you chose to Think when they are Leader, you may not be able to Think before your turn. Pay attention to how many cards they are holding and gauge their intent.

6) Control the endgame
The VP lead changes hands frequently and rapidly in GtR, so try to steer the game to end when you're on top. A mostly finished Catacomb, a Sewer, or a large Craftsman/Architect retinue give you significant control over when the game finishes. "Man, just one more round and I would have won!", is a common post-mortem refrain; avoid having to sing that tune. The Forum victory is difficult to actually achieve, but using the Forum as a threat to limit your opponents' actions can be immensely powerful. Also, you'll sometimes hit a score plateau for myriad reasons, like a limited vault, dry pool and stockpile, etc., where you need to change directions to make further progress. Keep an eye out for this effect and try to end the game before several rounds pass without your scoring any points.