This was definitely a sore spot for me too - everyone in the field always just dismisses the existence theory as standard and uninteresting, but for something so "standard" it's very hard to find a reference that actually applies! (The number of times I remember seeing people just cite Gilbarg & Trudinger for a result about parabolic equations on manifolds is disturbing.)
I guess we shouldn't be too surprised, though, when we consider the nature of PDE as a field of study: it's a smorgasbord of specific techniques rather than a pyramid of big theorems, and much of the work of PDE analysts is working out which techniques can be adapted to their problem (or perhaps more often: which problems their favoured techniques can attack!).
Anyway, here are the takeaways I had after a few years:
- Liebermann's book on parabolic equations is probably the most comprehensive you'll find. Like Thomas in the comments, I found it very difficult to follow; but it does have all the details you could possibly need for the case of Euclidean domains. One tip is to read it in tandem with Gilbarg & Trudinger - it's written as a "parabolic companion" to G&T, and follows its structure closely; so if you're having trouble understanding a proof, it's usually possible to go directly to its elliptic analogue and start there.
- Transferring existence & uniqueness from the Euclidean setting to manifolds is pretty technical. I found Charlie Baker's thesis on mean curvature flow to be a thorough reference for the case of closed manifolds. He proves a general existence theorem for non-linear parabolic systems which should give you short-time existence for Ricci-deTurck. (In general, these kinds of "uninteresting" gaps in the literature are often tackled in PhD theses, so they can be a fruitful place to look for details other texts see as a waste of space.)
As for regularity, the solution you get out of a short-time existence theorem will be as smooth as the initial data allows, but with norms that may (a priori) blow up as you approach the "final" time. The estimates required to rule out (or characterize) this blowup are usually intimately tied to the geometry, so you're much more likely to find them proved in geometric analysis texts.