Pro-ject have a very good reputation. Rega are also very good but maybe more expensive. At that price Pro-ject will do fine.
If the amp covers the CD player to speakers bit then it’s an integrated amp (i.e. both pre-amp and power amp in the same box).
Does the amp have a line in called “phono”? If yes, job’s a goodun, plug it in and you’re away. If not, you need a phono stage because turntables are quieter than other sources. There are plenty of phono stages about but don’t pay too much given the rest of the system. Google is your friend here.
Same advice probably applies to the turntable. Don’t bother spending too much unless you’re going to upgrade the whole lot. Better to spend the money on new music.