I never use primer as most of my pipe and fittings are less than a year or two old. If the pipe is clean and somewhat new so that there is not a layer of oxidation on it, no primer needed. you can pretty much do the joints all at once IF you prep them first. Do get some medium fine sandpaper and do sand all the parts which join each other. inside the fittings as well as the outside of the pipe. This gives a place for the cement to begin softening the PVC. It is a weld, not a glue joint in that they do melt together. Sand, test fit, apply glue quickly to both parts and slap together quickly. Hold for like 20 seconds if the pipe is more than 1" or so. They can begin to work back out if you let them go too quick.
Rather than worry about leaks, I find it more trouble to make sure that the elbows are pointed in just the right direction.
For a giggle, take a cheap fitting like an el, glue it on a pipe and wait 30 seconds, then try to get it off? If you are working faster than the cement, you're a whiz!