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Just finished my first electric and now am at the point of intonating it. Can someone describe the process for doing so? I assume I get each open string in tune and then check if it is in tune at the 12th fret. Anything else I should know or be doing?

Lots of things have to be checked but if you get the first three right then the rest tends to fall onto place. Yes open and 12 fretted need to on pitch with each other with in a given tolerance but so does open and 12 harmonic. Open and 12th harmonic are natural scale length notes one is full length and the other is at the half scale node. These two notes impart no stretch to the string out side of oscillation of the string. Because of this these two are the easiest to set to first. The 12 fretted note will most likely be a tad sharp because the string stretches as it is fretted. Add to that finger placement can and will affect how much it stretches this note has a tolerance limit built in just by its self so use a capo fairly near the fret and at the same place each time to be consistent. If you can get all three of these notes in tune equally with in a cent or two you are in good shape. you will never get all three perfect so don't drive yourself made trying. A guitar is not a perfect pitch instrument it is a tempered pith instrument.

Make sure that your Nut slots are at correct depth and that the neck relief and bridge saddle height are set to your preferences first and use fresh strings. All these will affect the innotation so get them right first.

I will disagree with Michael's comments above, in that I never bother checking the stopped harmonics, it is just not relevant.

Tune the open strings, fret at 12th with normal pressure not a death grip and check the note produced, if sharp move the saddle back(away from the neck). retune open string check again, etc till open and 12th are both right.

I've tried the 12th fret harmonic, but prefer fretting at the 12th.

Set up action, nut and relief first, then intonate.Do them in playing position for best results.
Gravity has a way of changing the relief ,especially in lightly built guitars.

I never felt the need for a strobe tuner. To expensive for me. If you already have one fine but i have done it for years with a korg chromatic tuner using the 12th fret method. And like Brett mentioned do it in a playing position. Sounds akward but it really isn't.