The biggest cook-pot I have available to mash my grain and boil the wort is about 3 gallons, but I'm trying to make a 5 gallon recipe (got the 5-gal carboy and plastic fermenter, etc). Can I make that work, or do I have to do the cooking in something that will hold 5 gallons?
You can do a partial-volume boil, diluting the boil volume with sanitary water. In fact, this is a great way to chill the batch. Your hop utilization in particular will be affected by a partial-volume boil.
You'll probably only be able to boil 2 or 2.5gl in a 3gl pot. And in any case, watch out for the very likely, and very messy, boilover.