You have several options each giving different routes in the future.
Turntable: with built in phono pre amp means you can plug into aux inputs on an amplifier for passive speakers or into most powered and active speakers with the right cable.
Turntabe with Bluetooth might sound great now but would grate when you find the limitation of Bluetooth (compressed, lossy).
In the 70s and 80s the mantra was to spend the most of your budget on the turntable or source.
Desktop or stand mount speakers are smaller, but if on stands may take up the same space as floor standing. Some stand mount speakers offer more bass than a similar priced floor standing speaker so check the specification and listen yourself.
There are wireless solutions but the turntable is likely to need wiring into a hub or amplifier (with passive speakers) or into one of the powered or active speakers (they have built in amplifiers which means the engineer designed the best match of electronics to speakers and with fewer boxes and power supplies better performance for the money).
The gadget folk like separates with rack of electronics and lots of cables. Families may prefer less boxes e.g. turntable and powered or active speakers (no rack of electronics but perhaps a phono stage pre-amp. Built in phono stage pre-amp in the turntable or powered speakers are not usually very flexible or good compared to stand alone options from Rega or Pro-ject etc. If you have separate arm, cartridge, turntable, phono-pre-amp, preamp, power amp and passive speakers you can tweak forever to get a better sound upgrade heaven or hell chasing the hi-fi dragon ...
Consider a simple set up leaving a lot of cash to buy albums.
Ruark MR1 MKII Speakers with Pro-Ject E1 Phono
Go upscale with powered speakers to:-
Q Acoutics M20
Triangle LN01A
Wharfedale Diamond A1
Klipsch The Fives
Triangle AIO
outside the budget when adding turntable:
SVS Prime Wireless Pro
Kef LSX II LT
DALI Oberon 1C with hub
and up the ranges of Pro-ject or Rega turntables. Rega Planar 1 perhaps?
If going the standard stereo turntable, amp and passive speakers there are thousands of more choices and a little harder in having to match the amplifier to the speaker or expect to have a nagging feeling you could have picked better. Very likely but really better starts at doubling your costs.
An Audiolab 6000A Integrated Amplifier second hand or Cambridge Audio AXA35 with Q ACOUSTICS 3030i Pro-ject E1 or Pro-Ject T1 EVO.
Try to get a demonstration. The look (you and family have to live with it) and sound quality has to satisfy you. Try to also listen to higher quality units so you can appreciate what is possible which helps also with deciding is half your budget gets close enough for your needs.