Just like the subject says, Im trying to have my drilled holes all have an exposed pad that links to the ground plane (from both sides). Can I do that or should I use vias instead? 
Yes, if you care about passing a signal through the board, a via is definitely the correct choice.
A via will not be exposed since it will be covered with soldermask by default. You could expose it by altering the solder mask layer above it.
An easier way would be to use a through hole pad since the soldermask layer is already adjusted so you can solder to it from both sides of the board.
-Bill
Vias are not covered with soldermask by default in Eagle at least. You would need to go into DRC and set a minimum size below which the via is automatically tented.
The other advantage of vias for this, again in Eagle, is that you don’t need to add the pad as a part in the schematic. The via is simply considered part of whatever signal you name it. Also, you can arbitrarily set the via drill size and ring size to get the exact effect you need (big drill with small exposed ring around it, square pad, etc).
A last option that would probably work in most PCB software: create a drill, then place a mask keepout on the top and bottom layers. If a hole is surrounded by copper but no soldermask, then it will automatically be connected through due to the way most PCBs are made.
macegr:
Vias are not covered with soldermask by default in Eagle at least.
Ahh yes, you’re right. I adjusted my DRC settings years ago for tented vias and haven’t thought about it since.
-Bill