What are the best option to Rekey My New House?
Rekey
You can probably rekey the locks yourself, depending on what brand and type of locks they are. The process isn't too difficult, you'll be a pro after one or two.
You'll have to remove the handles and deadbolts from the doors, so you'll probably want to go one at a time.  But if you like doing little projects like this, rekeying them yourself could save you some money.
Check amazon for rekey kits, but make sure you know what brand locks you'll be working on first.  I recently picked up a kit for less than $20. If I remember correctly, the kit had 50 of each pin and all the tools that were required.
If you decide to go this route, there are tons of great videos on YouTube that explain exactly how to do it.  Watch a few, and you'll be an expert in no time.
Locksmith
A locksmith will probably save you a bit over a wholesale change out. Again you'll probably want to do it in stages, unless the locksmith can guarantee they will be done in a couple hours.
Replacement
Replacing all the hardware is probably going to be the most expensive option.  However, it does give you the opportunity to switch brands, styles, quality, security level, and technology level.
Depending on how many doors you have, you might still have to rekey some of the new hardware (or special order it). Since most packages off the shelf only handle up to two doors.