You’ll need to have an instrument to practise on at home as soon as lessons begin (or regular access to an instrument elsewhere).
Here’s a very brief guide:
A well-maintained instrument is always going to be the best quality investment option. Full number of weighted keys, pedals and a proper piano sound. Holds value, looks great. But needs tuning at least annually and remember you might need to pay moving costs of £50 upwards. Budget at least £2000 for new, but used instruments can be great. I have a Schimmel 120 at home, which everyone loves!
A quality digital piano with fully weighted keys, and ideally a sturdy furniture-style casing is often a good alternative option. No ongoing-maintenance costs, usually a few extra features, like headphones for silent practice, and in-build metronomes. Budget in the region of £300 upwards for a new one. (Look at Yamaha Arius, YDP200, Casio Privia and Korg B2SP) If buying second hand try to go for one as new as you can, as the technology improves every year.
If you want to try out lessons before spending lots on an instrument, then you could use a keyboard temporarily but realistically only for a term before you’ll need to upgrade, so do plan for this!
(Keyboards are an accessible budget option and have fun sounds. They are better than nothing! But they don’t have the right number of keys, have a completely different weight, and usually don’t have a pedal. This can be really confusing and frustrating for pupils, derails good technique, and ultimately can put them off if used for too long).
Expect to pay £50 to £100 new, get as many fully-szied keys as you can (67 is pretty standard).
I am always happy to advise new and prospective students on instrument options!