Regulations on this will differ by state. The best move would be to find a local real estate lawyer. They wil instruct you on how to file all the proper paperwork with the local government, and also advise you on things you haven’t thought about, like how to structure your wills to make sure everyone is protected in the event something unfortunate happens to either of you.
Also don’t use this private transaction as an excuse to save money to skip any steps. Go get the title search and inspection done. You don’t want any surprises. If the inspection finds a fundamental problem, you might still go through with the purchase, but at least you’ll know up front what you’re in for.
In addition to all of the very good advice in this thread, I will add I am a big fan of Dollar Cost Averaging. If you have a large amount to put into the market, don’t put it in to whatever fund you decide all at once, put it in on a monthly basis. This protects you, to some extent, from the market taking a dump the day after you buy, because you are always buying. And your cost at the end is an average of all the times you bought in, and is not so much tied to prices on the day you bought in.
This may involve some planning for moving money around, because you will want to keep the remainder in a good HYSA in the meantime.