R.J.
We moved around a lot when kiddo was little, and for us, it was mostly apts (in some areas, rental houses are cheaper than apts).
We ALWAYS lived on the bottom floor... so we could jump, walk, run, bounce, slide, and ride (indoor rideable toys, like the PlasmaCar -indoor/outdoot, are awesome!) to our hearts' content.
ALSO... don't forget to look into low-income or "income verified" (meaning you have to make below a certain number) housing. YEARS later I'm still kicking myself, because just two blocks down from us (aka same neighborhood) there was a townhome complex that was low income / "income verified". SWEET townhomes with yards, and playgrounds, and just over all amazing. Filled mostly with single parents, student parents, and young families just starting out. All for 25% of what we were paying for our MUCH smaller, no yard, apt. And we qualified by a mile. Sigh.
When you're looking for apts... LOOK for child-friendly apts. Ones with playgrounds that are in use AND have parents present, kids toys left about (means that it's safe to leave them about/ won't get stolen), and some even have onsite daycare or YMCA pickup. When you're in a community that is mostly family oriented "kid noise" is just that, no big deal.
ALSO look for apts with washer/dryers in unit, and be sure to check out the heating/air conditioning. Which are BIG when you're in a small space with a small child.