What would be a good way to get wifi there?

there are many devices for amplifying/extending the signal

don't get anything unless you really have to....if your WIFI router has dual bands and is less than 2 years old, the signal will go a long way. Try it out in the new house first.

your results could be surprising

What Router are you using? What Bands. What is the constrcuction of the interior walls. Things like plaster/wire mesh act like faraday cage.

What I do fo good coverage in my house. is firewall in basement. then Time Capsule in the Office(bridged) and Airport extreme upstairs (bridged) Same SSID, Different Channels. but I also have Cat-5 in walls

they do make range extenders/repeaters. but YMMV depending upon lots of factors. but as scaryoldguy says, try it first and see what signal level looks like

While the walls are open, I was wondering if I should string a wire from the "office" to the bedroom or living room, and, if so, what type of wire.

Thank you

if you think you can crimp RJ45s on both ends correctly, do out! (or purchase long ones that are complete) - well worth the speed and security!

making CAT5 cables is not difficult - if you have the parts available you can network the whole pace!

yes the walls are open, i would run minimum CAT-5E everywhere you can. Or better yet, interduct with pull strings . If going in wall, get riser cable. Crimping / punching it down really isn't hard.

if doing a limited number of runs, you can get cat5e jacks at lowes/home depot that snap into face plate in a single gang box.

then just need to run cable and follow instructions/color codes for punching it down. you can run GIG-E 100M, and no worries about interfence (Well don't run the wall cable inline/phase with AC power).

Thank you so much. I have a Cisco Linksys E4200 router. It's about 3+ years old, but is dual band. I am thinking of a new router when I move.

If you are staying with wireless, look at brands/models that have matching booster/companion boxes so you'll have the easiest time.