I'm usually quite eager on using technology wherever sensibly applicable, however whenever I make some calculations I still end up using a pen and paper, by now resulting in an entire pile of sheets with 50% crossed out and obviously lacking any searchability. Every now and then I try to re-order this stuff by LaTeX-ing it, and sometimes I even try starting the calculations directly by typesetting it (and removing mistakes on the fly), but this seems pretty tedious.
So, I'm trying to come up with a question here that is not too subjective or broad, and it boils down on the question's title. But I'll try to elaborate a bit more:
- Where to get started using a computer/laptop/tablet/smartphone properly for back-of-the-envelope calculations? LaTeX? Mathematica/Maple?
- A keyboard may be a swift input method for text, but for formula I still consider my handwriting superior. That is however inversely proportional to its readability... So while there is a lot of handwriting-recognition software around, is there anything actually working well with formula? (Ideally capable of recognizing the most basic mistakes like sign errors, but I guess I'm wish-thinking there...)
Let's your two problems separately.
While I often use computer algebra systems (Mathematica, Maxima, Sage, Wolfram Alpha) to work out thorny integrals and complicated derivatives, I've never seen one that will assist with line-by-line derivations involving a lot of symbolic manipulation. For this, I always use pen and paper. As you said, typing equations is a huge hindrance to working with equations. The way I work is too free-form with a lot of cross outs, arrows, and side calculations. Standard mathematical writing evolved to be conducive to mathematical thinking, not typing.
As for catching sign errors and the like, I can only recommend proofreading and rewriting the results in a cleaned up fashion, similar to what you'd do with any other piece of writing. It does get annoying when I find that an $x/y$ turns into a $y/x$ after turning a page. Often, when I'm rewriting equations for publication, I'll catch these kinds of errors since the rewritten page is a lot cleaner. I'm also lucky to be a physicist so I can catch a lot of errors my making sure units still make sense every few steps.
LaTeX is definitely not the way to go when you're doing math. LaTeX is for preparing finished math for publication.
I usually work in bound notebooks, and if I get a result that I want to remember, I'll mark the location in a notebook with a sticky note. A better method would be to get a bound notebook with numbered pages and use the first few pages as a table of contents for interesting results.
If you want computerized, LaTeX is one way to go, depending on if you already know it. Microsoft Word has vastly improved its equation editor at least since version 2010, so it's actually rather nice to use that for recording results.