I wouldn't recommend a heated barn.
too much of a fire hazard. For the horse, free choice hay and perhaps a blanket will keep him plenty warm in the winter. As for people, if you're working hard enough you'll stay warm!
In terms of climate control, the only place that might need that is a tack room, and that need can be met with a dehumidifier or humidifier. Again, for the fire risk, I would not recommend an unattended space heater or milkhouse heater. I just keep leather tack in the house. You should have a fire extinguisher rated for the type of fires that could arise, keep it handy, and know how to use it.
Also, I highly recommend installing GFI outlets. They cost a little more, but are well worth the safety, especially if you plan to use clippers or electric water buckets. The GFI plugs will "pop" and stop working if they short out, rather than risking electrocution.
A further electrical concern is the lighting - if there are light fixtures inside the stall, be sure that the bulb is encased in such a way that the horse cannot rear and break it, or any broken glass will be prevented from falling into the feed, etc. I just have fixtures outside the stalls, but a farm supply store should be able to guide you to some livestock-safe light fixtures. In terms of basic startup requirements, you'll need... For the stall:
water bucket and mounting hook, feed bin/manger and mounting hardware, salt/mineral blocks (there are holders for these, or you could rough it), and a hayrack (optional - my guys eat off the floor in a corner where I deliberately don't spread bedding), rubber floor mats, and bedding (sawdust, shavings, straw, pelletized bedding, paper.... the list goes on - research what works best for your budget, availability, and needs). Grooming Kit:
Shedding blade, metal currycomb (for cleaning brushes), rubber currycomb, stiff brush, medium brush, soft brush, mane/tail hairbrush, hoof pick, and a tote or bucket to store it all in. For the Barn:
Manure fork and muck bucket/wheelbarrow, pitchfork, flat-edged shovel, tie rings, tack hooks/racks, feed storage (galvanized steel trash cans with bungee cords holding the lids tight are wonderful - mice chew right through plastic), grain/hay scale, fly repellent.
Further considerations would be: adequate turnout with secure and horse-friendly fencing, access to water for drinking and cleaning/bathing, manure disposal (spreader or compost pile are the most common), and vehicle access for trailers, pickup trucks, etc.
That's what I can think of right now... I'll pop back in if anything else crucial comes to mind. Congrats on your restoration project, and on bringing your horse home! It's a lot more work than boarding, but it is very much worth it.