How long should I keep a cut onion? Since I only cook for one or two, I will tend to buy an onion and only use part of it, and then keep the rest in the fridge. Sometimes it will be a while before I use the rest. Is there any guidance on when I should finally just throw it out? What are the signs to look for? What degradations should I expect to trade off against constantly throwing out half an onion (decreased flavor, increased bitterness)?
When I use fresh onions, we store the unused parts for up to a week in our fridge in either a ziploc or a sealable rubbermaid-style container.  For particularly pungent onions they go in the crisper drawer to keep the smell from being too strong in the rest of the fridge, but usually they're just on one of the shelves.  Stilltasty says 2-3 days but my personal experience says otherwise.

Generally in our house, they don't last long enough for the flavor to degrade too much, but they will get a little dried and the smell will get stronger over time.  After a week, they shouldn't be bad in the sense that you'll get sick - but you'll want to use them in a meal where the onion is the star, as the aroma will take over the dish.

Another option that I'm a big fan of is to dice or strip the leftovers (or, a 5 lb bag at once!) and freeze them on a cookie sheet (instead of in a bag, where they'll turn into a giant clump).  Once fully frozen, dump them into a plastic freezer bag, and store in the freezer for easy access to a handful at a time.  For most applications, you can defrost them right in the pan/pot/casserole as you cook, so they're very handy and accessible.  We do the same with peppers in our house for convenience.