looking for a vinyl player that also plays cds

The only thing approaching Hi-Fi that will do both in a neat package on this kind of budget is an early '80s mini or midi system from back when they were well made,  something like the Sony FH-7 mini system with a matching PS-Q7 turntable and a D-5(0) PCDP.

http://www.pocketcalculatorshow.com/boombox/decline4.html

The turntable even has a built in headphone amp!

The systems I mention like the Sony FH-7 are early attempts at miniaturising components into usually more compact visually matching systems. The best ones date to around the time CD was introduced or just before and will generally have a pre / power amp or component PSU and then cassette and radio. Here is a good review http://www.hi-fiworld.co.uk/hfw/oldeworldehtml/aiwaimicrosys.html

The Sony FH-7 system has a matching record deck which is actually around 1/4 the size of a 12" single but is very good  http://www.youtube.com/watch?v=PiG2F_tHeQ4. These were the days of the Sony Walkman remember.

There are more conventionally sized ones like th PS-FL77 http://www.vinylengine.com/library/sony/ps-fl77.shtml which also has a matching system.

The early Sony or Philips CD players would be the best match visually being around the same profile and these were so well made they pretty much go on for ever.

A lot of this kind of vintage stuff is becomming very collectible, but these midi systems are often overlooked as what came later when they hit mass market in the late '80s went steadily downhill. A bit like the then ubiquitous direct drive turntable.

Like Erik says you'll find it much easier to get hold of good quality 2nd hand components which won't often visually match but if your main criteria is audio performance then this shouldn't worry you too much.

A reciever is an amplifier with a radio built in, generally with a phono stage for a turntable connection and several tape loops. Often they have decent headphone amps too. Pioneer and Marantz are very popular but all the big Japanese brands made them so they are easy enough to find. They usually look like this http://www.youtube.com/watch?v=2q-3Uta5xTM

Here is a thread I made a few years back about the kind of turntables to look for on your budget. http://www.head-fi.org/forum/thread/228059/low-budget-vinyl-source

2nd hand CD players a pretty easy to pick up and inexpensive as many people are now ditching them in favour of hard disc based systems. A good separate component CD is still about as good as digital audio gets though for pin money.