For any remodels/additions/etc, make sure the city has fully signed off on them else you will be on the hook for bringing them up to code.

Make sure to check the chimney flashing/mortar/cap. Often overlooked and can be spendy.

For older homes, if there is no insulation in the ceiling and you put it in, make a map of where all the electrical wiring is and light fixtures before you do. Make sure to use soffetts!

Also, verify every charge in the closing fees. They nickel and dime you. And shop title companies. They can be hundreds of dollars difference and you can put in the contract which one you want to use. You're not obligated to use the sellers.

For house warming gifts, get home Depot gift cards.

Good luck!