Built in speakers isn't common on decent modern equipment, maybe some vintage record players, as apposed to turntables, and of course the cabinet models. There are small (slightly larger than a vhs tape) audio amplifies with preamps, that could output to small cd player type speakers you may have already or giant floor standers, though that doesn't sound like what you want. Do check reviews on amazon.
Are you at all interested in playing 78Rpm? My grandfather has allot of country from his on 78Rpm, such as Tennessee Ernie ford, some of it may be on 33/45, allot didn't get transferred as the genre changed just like allot music didn't make it from 33RPM to CD. Anyway I've found the desire to play 78RPM rather limits the options as far as new players are concerned. They are available but not near as many, so suggestions may be easier as options are fewer(if you insist on new).
If you can spot one at a yard sale or if thrift shops have decent prices on components in your area, you may check that out, I picked my player up from a yard sale for about $25, I later found that it seems to sell for about $120 on eBay, check the yard sales. Working with a small budget, I'd ask that you check your surround sound system, if you have one, mine has a phono input, so it can serve as the amp for my turntable, perhaps you'res does as well.
Dinner music you say? That's what 16RPM was for, not that I or likely anyone else will recommend that due to low audio quality and strange/rare parts, and limited selection.
If you're wanting somewhat continuous music you may look into a older record changer (don't know of any modern examples), they have a spindle that drops the next record on the stack when the previous is played, they were especially common with 45RPM. From what I hear the properties are quite large and land is cheap in the south, at least compared to the west coast, so if you have the room, a cabinet type record player may be exactly what you're looking for, and most don't want them because of how much room they take so they tend to show up for sale cheap, however they provided space for storing records, some had decent players built in, tended to be record changers, also they had a radio, and built in speaker(s). Some are very nice looking, and they tend to be 4-speed so they can play just about any record you can find. I've seen some selling for $30, if you can find one locally to avoid the shipping fees, it sounds like what you may be looking for. They also tend to have ceramic flip over needles which are convenient if you have allot of 78rpm as well as 33, however I've read that many of them can be refitted a newer magnetic cartridge as more or less drop in replacements to improve the sound, not that I've heard a cabinet with sound so bad as to insist on finding a way to improve it. You can always use a mp3 player fm transmitter for a car to make any cabinet table become a giant wireless speaker for a phone/other, I only bring this side note up as some look to junk them thinking they are only good for playing records.
One thing to consider with an old player however is that idler wheels on idler drive tables can wear out and I've seen some replacements at the $100 range, so check if it's an idler drive and like any belt or idler drive table if the typical maintenance parts are readily available. Some idler drives were especially bad but like anything there were good and bad models. As far as newer stuff, make sure whatever you consider does not have the pfanstiehl p-188 cartridge which is what the crossly and it's army of clones use, their arms make it onto non-clones occasionally as well.
Perhaps if you can find a used crosley type that properly maintains speed, you could spend about $30 to get a decent modern tone arm for it, again using cd type portable speakers, or put said modern arm on a cabinet table, as the removable universal head shells would make some old cabinets vastly easier to replace needles in, although you would likely loose the record changer functionality on the cabinet.