I would agree that a 5-wt. is the best choice for an all-around trout rod, which is ironic, given that I don't own one. No one rod is ideal for all conditions and situations, but the 5-wt., being very much in the middle of the range, will give you options to keep you fishing just about everywhere. You will probably wish you had something lighter for fishing the mountain streams, but a 3 or 4-wt. can be useless in a stiff wind or when you need to cast large, weighted flies long distances (the lake comes to mind), so I say that sacrificing a bit of finesse in favor of a more versatile backbone would be your best bet.

Personally, my first and only rod for years (when one rod was all I could afford, and only marginally so at that) was a 6-wt. I bought a 6-wt. so that I would be able to fish trout and steelhead with the same rod. It worked for both (and more), but it was perfect for neither. These days, I use a 4-wt. for almost all my trout fishing and an 8-wt. for salmon and steelhead. The 6-wt. has its sweet spots (pinks, large trout, and greased line summer steelhead), but it rarely comes off the shelf anymore.

For what you're doing, I think the 5 will give you the best versatility.

As for flies, take plenty of terrestrials (ants, beetles, and hoppers) for the lakes, and a few hairwings (caddis, stimulators, etc.) for the streams. Weighted streamers will work in both situations.

In other words, what these other guys said.