If you \usepackage[utf8x]{inputenc}, a lot of UTF-8 input will just work.
Greek is an exception though; it would always interpret e.g. α as \textalpha, which doesn't exist, instead of \alpha, which does. The only solution I could find was to write a simple package defining each of the \textgreek as \greek.
There are a few other exceptions (⨂ and ↦ come to mind), and I haven't worked out how to add support for a character (instead of just redefining it like I did with the greek). But the situation is mostly okay. I currently have this line in a LaTeX document for example, and it works fine:
Given $R$-modules $M,N$, their tensor product is an $R$-module $M ⊗_R N$ together with an $R$-bilinear map $b: M × N → M ⊗_R N$ satisfying: whenever $φ:M×N→P$ is bilinear, $∃!g:M⊗N→P$ linear such that $φ = g∘b$.
Sure, if you manage to enter the character XeTeX will be happy to typeset it. It's not a LaTeX issue, it's an editor/desktop issue. I don't really know about maths, but I'm guessing keyboard input for many special symbols is hard. I use (Xe)LaTeX to input phonetic symbols, and I hung onto tipa (the LaTeX package that gives you IPA from ASCII) for a long time, until I discovered C-x RET C-\ ipa-x-sampa.
You can also use TeX mode so that typing \phi will give you φ :).
Also, the very first time you use an input mode, you can just type C-\ to choose it. After you choose one C-\ toggles between that and the standard one, which is why you need C-x RET C-\ to choose another one instead.
How do you type Φ in LaTeX?