I have well water, and my pressure is pretty pathetic.

When I was pricing out an irrigation system last year, I also had to
include the cost of upgrading my well pump and tank.
However, the plumber explained to me that these upgrades didn't have
anything to do with *pressure*. They were required so that the pump
would run longer, instead of lots of cycling off and on, which would
wear it out the pump.

I tested my gpm at an outdoor faucet: 5 gpm! Terrible.

For an indoor faucet, that's not extremely bad. Indoor plumbing often
involves long runs of small-diameter pipe. Even with decent pressure, you
rarely (if ever) get enough water out of the indoor plumbing system to run
a sprinkling system.

My problem is, I want to get more radius out of my traveling sprinkler
(haven't commited to the cost of irrigation yet).

So I'm confused about pressure versus flow rate.
Will using a larger diameter hose do anything for me?

It depends on the length of the hose and flow rate. If your hose is
small enough in diameter, long enough in length, and you're trying to run
a decent amount of water through it, then the answer would be YES: The
pressure loss over a long, thin hose might be enough that a substantically
larger hose will help. However, if you're hooking the hose up to a spigot
connected to the indoor plumbing, then the answer is probably NO: The
indoor plumbing system probably wouldn't let enough water through anyway.

You could always test by using a short, fat hose and see if it makes a
difference....

steve