Intonation. Whaaaaa?

I hope I'm not violating any rules (spoken or unspoken) by posting this question, which may seem absurd to many, but is still confusing me. And I'd like to learn whether a) it's something I'm hearing, and b) can I fix it myself?

Here's the issue: I have a small travel/practice guitar that I suspect is suffering from an intonation problem. Except, I'm not quite sure what intonation refers to. When I tune the guitar (usually by ear, using string-to-string tuning), the guitar sound great. -- from about the fifth fret up. When I play open chords near the nut, the chords come out sounding "sour" (sorry, can't quite describe the notion; I played for my sister and BIL tonight, and they said everything sounded great).

My first thought is to replace the strings, and maybe go to a slightly thicker gauge (I come from the acoustic world, so I believe I mash the strings more than I need to on an electric, which this travel guitar is). But I have no problem playing on the upper registers, the 5th, 7th and 12th string harmonics sound good, but pretty much everything played on the first three frets sounds "off."

From what I've read, this seems like it could be an intonation issue. The guitar came with a truss rod wrench, but no instructions on how (or when) to adjust it. Given all the above, I wouldn't be afraid to try my hand at adjusting the truss rod if I knew how to measure when it's correct or not.

Alternatively, I could take it to the luthier I've used for some of my other guitars, but it's a half day to do so, plus the $$$. And if I can do it myself on a cheap (<$500) guitar, why not?

Help? Information? Advice?

Thanks!

Very often, if a guitar intonates poorly on the first few frets, the problem is the nut. When you press down the string at the third fret, the string should just clear the first fret. If not, the action may be too high at the nut.

Even if the action is good, it does not rule out the nut. Poorly cut slots can cause intonation problems as well.

Still, it could be technique, especially if you're used to heavier strings.