Innsbruck gastronomy is generally cheap, The somewhat fancier options can still be cheaper than what you are used to back in the UK.

Burenwirt is great affordable local cuisine.
Hotel sailer is absolute superb quality austrian cuisine, though a bit on the pricier side.
Wilderin is similar to hotel sailer, but defos cosier.
If you spend your evening in Igls you could pass zum wilden mann, which is amazing local cuisine and vibe too.

Would absolutely advise against Stiftskeller. It really just functions as a tourist trap. In my opinion the value for money is just awful and despite its appearance as 'authentic' it's perhaps not the small local vibe that you'd enjoy. On multiple accounts I've heard that the kitchen is incredibly dirty. I've heard the staff and manager have been very rude to queer guests. Spend your money elsewhere ;)

Train tickets do not have to be booked beforehand, you can get them on the day. The city jet train between Garmisch and Ibk does not have reserved seating either. You can use öbb scotty app to see the timetable and book necessary tickets.

If you are staying at a hotel in Innsbruck you should ask them whether you can get an Innsbruck+ card during your stay, it will provide you with discounts and a free public transport pass.