Oh, how I understand the itch to get going!
It's totally doable. And I've gone on and on about it on my blog, so if you'd like, have a read.
The long and short of it is that with typical kids, it's way easier than you think. The fear and trepidation is way worse than the actuality.
As for a cheap idea, we're headed to Sayulita, Mexico this winter. It's a funky little town north of Puerto Vallarta.
We're renting a 1 bedroom suite in a small house for a month. It will work out to about $45/night. It has a modest kitchen, and a fenced yard. And no pool, which is a biggie for me while my two kids are not confident swimmers. We did the same in San Diego last winter.
As for sleeping, naps, etc ... the big trick is for you to chill out about it, which is way hard, I know. As for the safety piece, we've been known to haul king sized hotel mattresses onto the floor, shove beds against walls, play cards while baby napped in a questionable bed, etc. It can be done!
Happy travels!traveling with toddlerstraveling with babiestraveling with kids