I'm not sure what you mean by "couldn't get it up to 80." Would the air not go in at all, or did you keep pumping, hear the hiss as air went through the valve, but the pressure didn't rise, or what?
It's not likely both tires developed a problem simultaneously, so I suspect the pump or gauge. Many gauges are junk--they work inaccurately for awhile, then quit, or die, or read wrong.
as for the pump, if there's an obvious way to open yours (unscrew an end plug?), do that and pull the guts out. There will be a shaft and a leather or plastic washer-looking thing on the end that seals against the inside of the pump. Make sure that's not folded or dirty or worn out, put a LIGHT coating of grease on the part that contacts the pump cylinder and reinstall carefully, so it doesn't wrinkle.
Check the "chuck," too, the part that clamps or slides onto the valve. They have soft gaskets inside that can wear out, and sometimes the cam goes. May be repairable, may not--your bike shop or the manufacturer's web site can probably tell you (generally, cheap pumps are throwaways; you can order repair kits for more expensive ones. A few companies have sent me free parts when I've asked about buying them, or you can sometimes find hardware and hoses at places like Home Depot). In any case, pumps are pretty simple. I've made or substituted parts several times with materials from hardware stores.
If it's a tire leak, fix it, but find the cause so it doesn't happen again. A sticking rim strip or liner isn't likely to cause a flat, in my experience, but it can. You can avoid stickage by dusting the inside of the tire with talc or baby powder when you reinstall everything.
To check for leaks, pull the tubes out of the tires, inflate them a little, and if you don't see or hear anything obvious, dunk them in a bucket of water and watch for bubbles. Try to keep track of where the holes are in relation to the tire and rim, and check those areas carefully for thorns, protruding spoke ends or whatever so you don't have the same problem again tomorrow.
if they don't work right when you put them on the rim, they won't work right after any number of miles. They don't have to "seat" or anything.
One last tip: when you install a tire and tube, rotate the tire so its label is at the valve and facing to the right side of the bike. Then when you do get a puncture, you can tell by the location of the hole in the tube where the sharp thing came through the tire and be sure it's not still in there. Saves a few seconds to several minutes looking for a little tiny thorn.