How to setup an audio system for TV and other audio sources?
I would go with a decent quality AV (audio video ) unit. These are essentially integrated amplifiers that handle/pass through video signals as well. You can spend as little as $200 or as much as much as many thousands of dollars. Assuming you are on a budget i would look for a mid to high level USED one ( unless you can pop for a new one ). 

I scored a Denon AVR 3301ci on craigslist for $300 that was $1600 retail when new. 

The lower end units may not offer you all the connect-ability you may want. the mid the upper end units will have lots of inputs for dvd, games, aux, pc, phono etc. Some of the newer units on the market have been built with streaming and Digital audio converter included. 

Make sure you get one with a phono input unless you already have a phono stage ( pre amp ) for your turntable other wise you will not be able to run a turntable through it. 

I would join an audio video forum such as AVSForum , and do lots of reading and research before you make and investment. Now there a lot of audiophiles on some these sites who extol the benefits of expensive this or that but there are a lot folks just like you who started out with little knowledge and or money and will love to help you get started. 

Some good names in the AV unit arena are Denon, Yamaha, Pioneer. 

Go the  Receivers, Amps, and Processors forum write a question and ask about recommendations for AV units with all of the needs you are looking for and you will get loads of recommendations. 

You will need speakers and these units can run two in stereo or as many as 9 or 12 in surround. You can start with two and as you get more ad them. 

I love my Denon and it rocks with movies or my LP's. I will say that the operation manuals from denon leave a lot to be desired but there is usually a forum dedicated to each specific unit with help and advise for those in need.