The following is assuming you are playing TA. Most of it will apply to DL and DA, but if you don't recognise something, that is likely why.

The cultural buildings themselves aren't worth that much. There are race-specific exceptions (Korath Dark Influence, Thalan Brain Washing Center), but the vanilla embassy/cultural exchange are pretty worthless.

Honestly, the morale boost from the political capitol is worth more than its influence bonus, simply because influence is so local. Slap that thing on a morale bonus tile, and you have a planet that generates considerably more influence and income, due both to the boost from the building and - more importantly - from the higher population the planet can support.

You are correct in believing influence deep in your territory doesn't help you on the front. Having massive influence generating planets three sectors away won't noticably help you flip any enemy planets. If you are seriously intending to take on the universe one iKorx at a time, concentrate mostly on empire-wide bonuses and population. Most offensive cultural work will be done using starbases to break into an enemy area, flip the first few planets, then letting those planets do most of the work of flipping the rest of them.

When placing influence bases, keep in mind their effect decreases with distance *squared*. Ignore the ring, it doesn't indicate squat. No, the modules you place on your starbase do not increase the influence of your planets in the area, they increase the influence coming from the base.

Build the thing touching an enemy planet, directly between two planets is even better. Depending on what tech tree you have, you can get to 300+% bonus. I recommend stopping well before that, because half (or more) of the modules only give 5%, 7%, 10% bonuses. Concentrate on the higher yield modules and only use the low ones when you already have 4 bases in the sector. More often, building another base next to the first is a more efficient use of your constructors.

Once you get a planet above the magic 4x threshhold, you can lay off the bases. Getting a higher influence ratio does not improve your chances of flipping the planet. Get the Mind Control Center (in TA that is, avoid it like the plague in DL/DA). Seriously, for an influence-based conquest, this is an at-all-costs necessity. It flips any planet that gets above the 4x level the first turn possible, no more waiting 40 turns for the game to randomly flip it.

The Restaurant of Eternity and influence resources are worth wars over. There is no excuse for allowing anyone to build the RoE before you; if necessary, rush buy your empire into insolvency for a few turns. If for some reason you fail at this, take it as soon as possible from whoever has it. The same goes for influence resources. If you lack the capacity to take on whoever has it, pay someone to declare war and hope they destroy the offending mine. Just make sure you have constructors waiting to take it when they do.

If you have further questions, speak up. Nothing's more fun than passing on the true way to a new recruit   

Just to toss 2bc in here, the MCC is great in DL/DA if you don't mind not flipping planets.  It's (bugged) 100% econ bonus empire-wide is a very big boon otherwise.  I understand this thread is about culture, butI just didn't want anybody to think the MCC in DL/DA wasn't worth a planetary spot.

As for the rest of the post, Willthemailboy is THE MAN for culture around here, so anything on that subject you hear from him should be golden.

I'm new around here, where would you recommend I go to see the local art and history of the area.  Any good cultural restaurants to sample the food?  I here its to die for...

No, it does not get the bonus from a tile. Buildings that affect your entire empire do not ever gain bonuses from tiles. This includes RoE and all trade goods.

Planetary buildings come in two classes. Class one is manufacturing, research, and farm structures. Buildings in class one can use tile bonuses only if they directly produce manufacturing, research, or food points. They do NOT use tile bonuses if they improve the output of related buildings (power plants, research coordination, food distribution center, etc). If it says adds x% to production, don't put it on a bonus.

Class two is morale and influence buildings, since these buildings cannot not directly produce points. Buildings like this DO use tile bonuses when they add x% to morale/influence. This means the political capitol is the only one that should be on a bonus tile, and it can be on either morale or influence - it can use both.

how do we get the racial symbols, can we get more than one?  how doyou achieve a racial victory?

The only time I ever build influence buildings is near the beginning of the game. During the colony rush, I never settle on planet 7 or lower (rarely, unless I need it for some other reason), due to the fact they are too expensive to be worth having until later in the game. When other civs start taking those planets, I will use influence buildings to eventually flip those planets, then get rid of the influence buildings for something more helpful. Although using influence starbases is probably better than wasting planet space (unless my planet may flip).