Bicycle Mechanics - used trek 830 where do i start

hi every one new to cycling bought a used trek 830 for $15 i do not know where to start the bike has rust on the chain and when u back pedel the chain get loose the gear shifting is very noisy and sloppy how do you clean rust off off the chain and some part of the wheels

I would just get a new chain. It looks like a 7-speed there and a chain for those are around $10 (http://www.nashbar.com/profile.cfm?category=&subcategory=&brand=&sku=12652&storetype=&estoreid=&pagename=Show%20All%20Products).
I would take apart the cassette and use some naval jelly to get rid of the rust. Or just buy a new cassette, here is one for $12 (http://www.nashbar.com/profile.cfm?category=&subcategory=&brand=&sku=16451&storetype=&estoreid=&pagename=Show%20All%20Products)

I realize you only bought the bike for $15 but it is just going to take too long to get that rust off to save a few dollars.

Dump the chain and get a new one. Around $20. Try some WD40 on the rusted areas and lube your shifters with something like tri-flow lube.

I would either

(1) thoroughly clean the cassette and chain and keep them both, assuming you are not skipping currently under heavy cranking

(2) replace both the chain and cassette. They tend to wear together and I have seen it where you replace just the chain and there is skipping on an old cassette.

You can get a new chain at X mart for about $6.

You probably need to clean/lube all the cables.
For DER adjustments etc., see-
http://www.parktool.com/repair/