What gets you the best value for money are often a second hand vintage produced during the late '70s to mid '80s but you always want to test those first or be at the very least make sure it's from a reputable seller before you buy a second hand model through the internet. Pioneer is very reputable brand, especially for budget turntables, the same goes for Rega from what i've heard. The PL-990 as suggested has been produced for many years now and still is a popular model for a reason. It has a built-in pre-amp as well so make sure you connect it to the AUX socket of the amp. I wouldn't recommend going with built-in speakers unless the speakers you have are very poor. I'm a sporadic turntable user myself and far from an expert, I recommend you to quickly create an account on
http://www.vinylengine.com/turntable_forum/ and ask these guys for suggestions. It's a very active and reputable forum so it shouldn't take long before your the replies start rolling in. Just make clear that you're on a budget and also make sure you give them the model of your amp and speakers. Pretty sure many of their members will recommend you a second hand vintage model as well.