I've been trying to get into Fallout 3 but seem to have troubles with my game freezing at inopportune times (is there an opportune time?).

I'm not sure if there is a time where it becomes inevitable or if there is some kind of memory leakage happening, or something else. I did look for and installed an Unofficial Patch mod, but have no other mods installed. I do have all the DLCs (I think).

The worst instance was in Megaton. I went into Moriarty's and the game froze. After getting Fallout 3 closed, I tried again. It froze again when I tried going into the first building (the water treatment plant) in Megaton. I had a couple of freezes before ever leaving the vault. The latest one is the one that hurt the most. On my way from one little village (where Super Mutants stole some people away) to get some mines for Moira, I'm running along the road. I came across a couple of mutants and killed them (barely). Almost made it to the minefield when the game froze again. It had been a long time since my last quicksave.

Is this a game where I have to quicksave before and after every encounter along with maybe every 5-15 minutes just to be safe? Or is there something wrong with my game?

That sounds like an awful lot of freezes to me. I get an occasional freeze but nothing on that scale. My problem with Fallout 3 is CTDs. I've gotten more CTds when playing this game than any other game I have ever played.

I never quicksave in a Bethesda game. Back in the Morrowind days it caused savegame corruption so I got out of the habit. I do make regular full saves, though. I make a full save about every ten minutes. I never know when Fallout 3 is going to crash.

You really need to provide more information about your hardware before anyone can make any suggestions. Post a DXdiag of your system so people have something to work with. There could a number of reasons why those freezes are happening.