If want sound so good that you can tell if the signal has been digitized then you have a hugely unrealistic budget. By a factor of at least 10x.
That aid these are older likely well-used recordings. I doubt they are the kind of vinyl the audiophile talk about. Vinyl records can be very much degraded by being played even just a few times on a cheap player. You will have to check. You might be lucky or not.
That you need are FOUR things.
1) A turn table that is outfitted with tone arm and cartridge. This what takes the signal of the record and converts it to an analog electrical signal.
2) A "phono preamplifier" you can buy a stand alone pre-amp but almost every stereo receiver has one built-in. You might even own one already if you have an "AV Receiver" connected to your TV set. This is NOT optional. Al records are made with an kind of compression and the phono preamp decompresses the signal
3) A "power amplifier" This makes the signal from the pre-amp bigger so it can drive speakers
4) A Pair of speakers,
OPTIONS:
(A) Many times #2 and #3 are built into the same box. Many time also with an FM radio receiver inside. Goes by names like "stereo receiver" or "integrated amplifier"
(B) Some times #3 and #4 are built into the same box. These are called "powered speakers" or "active speakers"
Please do note that you can NOT simply plug a pair of powered speakers into a turn table. I think this is what you were asking for.
I think a reasonable low-end budget might be close to $600. When you do go shopping no matter what your budget you get the best result if half the budget is in speakers. Divide the other half over the stereo receiver and speakers.
These are not bad speakers for the price:
http://www.amazon.com/Polk-Audio-Monitor-AM6025-A-Floorstanding/dp/B0002Z2416/ref=pd_cp_e_1
They still make inexpensive turn tables too:
http://www.crutchfield.com/p_033DP29F/Denon-DP-29F.html?tp=200
A generic receiver is likey good enough
http://www.crutchfield.com/p_580TX8255/Onkyo-TX-8255.html?tp=47041
The above would be a "decent" analog system, not great but likely better than what most people have today. If you are filling to move the signal to digital they currently make some "USB Turntables" th=ey are low quality but also inexpensive, and of course don't preserve an analog signal path.
I'm sure others will find other gear to recommend. I'd not recommending anything, just showing examples