Do you get to go home first to help wife move before reporting to first duty station?

Yes, you request leave/PTDY in route. All of your planning and scheduling will begin at some point during your training. Good luck!

Do you have any leave saved up yet? You could use that to go get your wife, or when you get to your first duty station, your unit should give you 10 Days Permissive TDY (you request this through your Chain of Command on a DA31 - Request for leave).

This is not chargeable leave, so if you don't have that much leave saved up you can go home & take care of moving/POV recovery/house hunting. If you're on active duty & reporting to your permanent assignment, your unit/base will give you your inprocessing paperwork when you arrive & they give you like 2 weeks to get everything completed (Finance, CIF, Housing, etc.) but it might only take a few days. You can probably schedule your leave and/or PTDY to go get your wife & get everything done. If you speak to your supervisor and you're back at work, they may even give you like a 4-Day pass to complete moving, unpacking, etc. Just be up front with everyone on your plans, needs, family... most Commands support families so they should work with you (especially because you're new).

Yes!

The way it worked for me back in 2005 was I finished AIT, went straight to Airborne school, went home for hometown recruiting, then went to first duty station. Once I was linked up with my unit I was given 10 days to move my family down to Fort Bragg. I only made it like 3 days in before we were sent down to New Orleans for Operation Katrina though.

Back in the day you did. I remember reporting to Ft Lewis WA. 9th Infantry Division. My 1SG asked me three questions, "is your family settled in? are your finances ok? do you have food in the fridge?" To all I said yes 1SG. He then said, good, get you stuff ready we're going to the field. I stood there thinking "Damn, why didn't I get that hospital assignment.

Your new unit should assign you a sponsor who can guide you through a painful process. The Army will move you, you don't have to do it yourself.
That said, get rid of the junk, keep the good stuff, and plan on what you need to have arriving first - pots, pans, clothes bedding etc. - Maybe it will all ship at once, but you still need to function.
If you plan on driving your wife to the new duty station, that's fine, but use permissive TDY carefully and methodically.
Find out how long shipping takes, then do the backwards math.