I have proofs that run on for up to 10 pages. To aid in readability, I would like to discuss my theorem, and the lemmas that support it, without the noise of these lengthy proofs. Can I move the proofs to my lemmas, theorem, etc., into an appendix? Or somewhere else more appropriate?
Can I move the proofs to my lemmas, theorem, etc., into an appendix?
Yes - this is often done in journal articles. You might want to put a sketch of why your claims should be believed in the main text, with a pointer to the full proofs in the appendix.
But, you have tagged this thesis, so it would be worth discussing this with your thesis advisor, as they may have a different opinion for your particular field or their particular preferences/style.
Personally I feel that the appendix is only a place for the busywork kind of proofs. Examples would be something like long, tedious, but unsurprising calculations, or a specific variant of a well known theorem you need, that can't be found anywhere but is immediately obvious from some simple modifications of the original proof. In short, only the stuff that everyone would believe to be true from the get go and that just needs to be there to form a foundation for something more important somewhere else in your paper. In short, the stuff that generally no-one reads.
Conversely, if there is any "novelty" in your proof, it should never be in the appendix.
That being said, there is still no need to put proofs directly after the statement. It is quite common to discuss a set of theorems, lemmata in one section and then only do the proofs in the following sections. If you look at research papers, you will often find a section called "Proof of Theorem ...", where the Theorem number is smaller than the number of the section.
But if you do so please take care to minimize confusion, i.e. clearly reference forward and backwards. If it is badly executed, this style can be extremely annoying, as one always has to hunt for the theorem statement or the proof.
If you have "ten page proofs", perhaps you should see if it makes sense to break them up into shorter lemmata that can be reused or at least help structure the mess. Some detailed, case by case checks, other tedious busywork or programs to do the above can be relegated to an appendix, but ask your advisor beforehand.