Need a bit of help with my very first guitar build ever. Bought a kit from a well reviewed builder/seller and upon inspection there don't appear to be any blaring defects on the guitar. HOWEVER, there is a blaring problem. Whenever I play open chords, or pretty much anything involving fretted notes and open strings, the guitar sounds horribly out of tune. I go back, check the tuning, and it's fine. Try playing another open chord, sounds horribly out of tune again.

On top of mostly being problems with open strings/chords and lower frets(pretty much 1-5) the biggest problem is with the highest three strings.

I am not sure what the problem might be, but I have a few ideas. I would feel more comfortable going forward with a bit of guidance, so if anyone has any ideas, I would really appreciate some help.
I don't want to be condescending, but did you set it up first? as in, intonation, string height, etc? Sounds like the strings might be too high or the bridge too close/ far from the nut.

In short, something makes the strings have the wrong length. Maybe the scale is wrong. You could try measuring from the nut to the twelfth fret and then doubling that and see if the bridge is anywhere near that mark.

Is it a bolt-on neck?
Just a few follow up details that I have left out in my haste to try to find a solution to my problems.

The bridge is a tune-o-matic style bridge, the tailpiece is just a standard stop bar tailpiece, and the neck is bolt-on.

I attempted to intonate the guitar, but found that no matter how far back I adjusted the bridge saddles, the octaves were still coming up sharp. On top of this, the action on the guitar is VERY high. Obnoxiously so in fact. I know this can have a big effect on intonaton, so I'm sure that has something to do with it. My only problem is that I have the bridge set as low as it can go, but the action is still ridiculously high. I have heard that adjusting the neck angle by adding shims under the neck joint can help with this, so that might be something I will consider going forward.

If any one else has any suggestions, tips, or even a link to a good tutorial(written or video) it would be VERY much appreciated.
Tuneomatic bridges usually require a neck angle, around 4 degrees I think. This obviously depends how high above the body the fretboard sits though.

Make sure the neck is straight and fix the angle. If the height at the nut is too high then that will have an effect on it too.
Just wanted to thank everyone for the suggestions. Sorry it's been a while since I've responded, have been quite busy so I didn't have a chance to work on the guitar til yesterday. Threw a couple shims in the neck joint and that pretty much solved everything. The action is beautiful, I was able to pretty easily fix Tim intonation problems and now I'm quite happy with it.

Thanks again to everyone who came to lend a hand.