I was hoping you guys might have some insight into what I should be looking to do for building a fence. I'd like to try to keep costs down if possible to put money towards other projects.

At my old house, I had a run of about 350 LF that was built using 8 inch round posts driven into the ground by a skid loader (probably 3 feet or so), coated wire mesh ran along the outside of the posts and 1x6x10s (three high) nailed over top. I had two gates (1) five foot and (1) ten foot that was two sections. The fence was 5 ft tall.

I'd like to build a fence at my new house but would prefer to DIY to save on the labor costs. I was considering just recreating what was done at my old house but was wondering if there were cost savings to be had by skipping the 1x6x10s while still maintaining the integrity of the fence itself.

If so, I'd just need to get the posts, materials to build two gates, measure out the run and spacing for the posts, drive them in, mount the wire, and hang the gates, correct?

What is the proper method to mount the wire and should that be done on the outside or inside of the posts?

Any help you can lend would be appreciated.

Of course, if that line of thinking is wrong, I'm all ears as to what I should be looking at.

Depends.  What kind of posts and was it chain link fence or welded wire?  You can make it as cheaper as sturdy as you want.