Found the problem. You have the govenors controlling production. Using the govenors when you're first learning the game isn't a bad thing, but letting them control production is not a good idea, IMO.
I don't have a problem with govenors per se, especially when you're first learning the game. There are levels to civ (I don't mean Cheiftain to Sid, either) that can be very intimidating and overwhelming to a new player. Using the govenors helps with that, and makes it more manageable. So for now, using the govenor to control happiness isn't necessarily a bad idea. I won't get into the whole use-govenors vs don't-use-govenor debate....but I do believe that later on, after you've got a pretty firm grasp on the game, you should micromange the cities yourself.
Okay, so how to fix your problem? It's easy. Open a city. Any city. Go ahead, I'll wait. No! Not that one!
Contact your govenor ('G' in the city view, or click on the little head above the food box) and find the 'Manage Production' line under the 'General' tab. Select 'No' from the left drop down box, and 'In All Cities' from the right one. Click on the box on the bottom that says "Default Setting for New Cities", and close it out. You'll then be able to produce your own buildings/units/wonders in the cities.
Like I said, govenors to control happiness isn't a bad thing when you're starting out the game. But I think you should control your own production. The Govenor has some pretty weird picks sometimes.
Oh, and I merged your posts. On the lower right side of your posts, next to the quote button, is an edit button. We would prefer to have you use this than to double/triple post. Just keeps things neater.
And Welcome to the Forums!