Wanderer,
We did the same exact thing a year ago, so welcome to the club!
Two years previous we decided to move (from Maryland) and in spring of '09 put our house on the market and started the process. We've now been here 13 months almost exactly, and couldn't be happier!
After we had a buyer for our house, we left the kids with family and came down for a weekend. We have 2 small children plus a dog & a cat (well, we lost the dog to cancer a couple of months ago), so our requirements were pretty strict, but we did it. We looked at probably 30 houses, but late Sunday afternoon we found one that fit our bill to a "T" -- cat and all!
We had visited a couple of times before so we knew generally where we wanted to be, but we looked at nearby towns as well, including Simpsonville, Mauldin, Taylors, and Greer. We ended up in a little enclave called Half Mile Lake, north of the city. We are very close to Paris Mountain state park, and Cherrydale (where we do most of our shopping). It's great for us, except for not having ANY businesses within walking distance.
Here are some points for you to ponder:
Most apartment complexes are pet unfriendly--we didn't even go there. But there are SO MANY houses for rent, why even limit yourself? You'll get more room and privacy, even a yard for the dogs. Find a house, you'll be glad you did! Rents are very reasonable. Sure, we had to pay an extra security deposit, but we are responsible and expect to get it back. NBD!
If you want to walk to downtown, the "hip" place to be is the area called North Main. It starts, I dunno, I guess around Park Ave or maybe Stone (I'm not local!), and goes all the way up to Rutherford Rd., extending east and west from Main St. (westward to 25, eastward...who knows? Wade Hampton?). From the intersection of Main and Park Ave it's a 10 minute stroll to downtown. That said, it's almost entirely residential except for the first couple of blocks of Main and the major outer arteries mentioned, so you'll be driving for most purposes.
If you want to be closer to a "hip strip" you want to check out "Augusta Rd area." (For some reason everyone calls it "Augusta Road" but it's actually Augusta St.) This is south and then east of downtown, and there's a lot to do there. Cafes, restaurants, antique shops, salons and so on, and both above and below the main drag are some really nice neighboorhoods. There's plenty walk to, but it's a bit of a hike to downtown from there.
If you really want to be "in the thick of things," there are lofts and condos right around the downtown area. Probably one of the pricier options, but I don't know your budget.
One thing we did, and I highly recommend you do the same, is to get a realtor to show you around. Many times they can get commission for rentals, so you're not wasting their time. We were lucky enough to be introduced to a great guy, who spend the better part of Saturday driving us around. I can recommend him highly--if you want his contact info let me know. Although he wasn't responsible for the house we ended up in, he earned our loyalty so when we're ready to buy, he's da man! Plus, we've become golf buddies and his daughter baby sits for us. (This is a great freakin town!)
If you like driving in traffic, try to find a place around Woodruff Road.
Hope this helps, let me know if you have any more questions.
Cass
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Santa Fe was beautiful when I visited on vacations, but living here is a different thing entirely.
Once you come up with more specific questions, post them, and many folks will chime in with helpful info. But just reading back thru the Greenville Spartanburg Area forum will also be time well-spent. 
