Jim Roseberry
I was just going to chime in and say the Catalina maple kits sound exceptionally good for the low cost. 

Thanks for mentioning!
I REAAALLY thought about getting one - but the family I mentioned above is willing to sell their kit for $350 - EVERYTHING (minus good cymbals or good cymbal hardware- the supplied stuff is the cheapest hardware and cymbal set I've ever seen, and only came with a hat and a terrible excuse for a crash). I think we'll save a lot of money getting this one - it's not too loud when played well, and it doesn't actually take up a lot of space compared to what we expected. So win-win! (if this works out :D )

What about an electronic kit?  That will allow you to edit the sound to suit the environment.

I was going to say I would look for a good used kits as they can be found for around $300. They will have garbage cymbals and possibly the lower end hardware but sometimes you get lucky. The quality of the hardware is what will make or break a cheap kit. The shells if in good shape might not be the best tone wood, but can be fixed up with new head's etc.
Most cheap kits will have about the same serviceable grade of hardware. Even if your not a drummer one should be able to look at it and see if the hardware is sturdy and will hold things in place. Nothing worse than tom mounts that sag and stands that fall over when you hit a crash...
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The basic kit is the Kick and the Toms. That's the bulk of a drum set that needs to match. It's also the least expensive over all when shopping used and lower end kits. You can spend equal or more on cymbals or a good snare. And that pert of the kit is not as critical to a good drum sound as one would believe. You can make almost any kick drum sound the way you want with the right heads, tuning, dampening and mike placement.  And hopefully your Church drummer is not going to be overly busy and using tom fills at every opportunity so Toms are not critical either. A lot of drummers I work with have only one rack tom. 
The Hi hat's and the Snare are really the heart of a good drum sound and that's where I would spend the money. 
The rest can all be upgraded piece by piece as funding and opportunities allow. I would certainly grab that kit if the hardware looks like it will hold up. 
 
In the studio I like 18" and 20" kicks best of all. More snap and less Boom. 
And just keep your eyes out for deals on name brand cymbals. A drummer I'm working with right now just scored a set of four AA's for $500 lightly used.  
 
I agree that Digital kits are not the best solution unless that kit is a Roland worth $4,000. 

Cactus Music
Most cheap kits will have about the same serviceable grade of hardware. Even if your not a drummer one should be able to look at it and see if the hardware is sturdy and will hold things in place. Nothing worse than tom mounts that sag and stands that fall over when you hit a crash...
. 
The basic kit is the Kick and the Toms. That's the bulk of a drum set that needs to match. It's also the least expensive over all when shopping used and lower end kits. You can spend equal or more on cymbals or a good snare. And that pert of the kit is not as critical to a good drum sound as one would believe. You can make almost any kick drum sound the way you want with the right heads, tuning, dampening and mike placement.  And hopefully your Church drummer is not going to be overly busy and using tom fills at every opportunity so Toms are not critical either. A lot of drummers I work with have only one rack tom. 
The Hi hat's and the Snare are really the heart of a good drum sound and that's where I would spend the money. 
The rest can all be upgraded piece by piece as funding and opportunities allow. I would certainly grab that kit if the hardware looks like it will hold up. 

Great info, Cactus! 
 
The biggest problem is that the only 'drummers' we've got are myself (Only been playing 1.5 years - don't have so great technique but I at least know when a kit sounds good, and can play a good selection of grooves!), and the 14y.o. kid who will be playing them mostly, so the idea of bringing along a very experienced drummer seems to be slim. 
 
In any case, I'll keep my eyes peeled for those deals. We've got a very decently-sized budget that was actually reserved for an entire kit, but if we can split that amount of money into JUST this used kit, new heads, cymbals + hardware, and *possibly* a new snare, we're golden as far as not needing to invest in all-new shells which would raise the cost considerably from what I've seen (although the snare we've got is surprisingly good, just needed a little tuning and now it sounds great to my ears).
 
That's not to say that it's is the greatest snare I've ever played , but I don't think it'd be worthwhile funnelling money into upgrading it now - not until a couple years down the road when we see its shortcomings, maybe. I realized we likely won't be using it for recording within the first year or two, and it won't be mic'ed in the hall in that time so we can get away with making it sound as good as possible just as it is 
 
This is really helpful info, guys! Keep it coming!!! 

If that $350.00 kit works for you then that is great!  You will have a decent budget for cymbals/hardware.
Be sure to get a comfortable seat!  It makes playing so much more fun.  Tuning is the most important thing.  Proper tuning can make a cheap kit sound good.  Improper tuning can make an expensive kit sound real bad.
 
If there are no drum experts in your church, branch out to local music stores (especially mom & pop stores where the customer service is likely better than the giant conglomerates. Sweetwater is one exception to this rule!).
 
Find someone who can help you learn to tune them.  It is an art and one that is VERY subjective.  If the local music store has some kits set up, find one you like the sound of and ask who tuned it.  Ask them for help showing you what they did and WHY!  It will go a long way towards getting the sound you want.  Personally I believe this one aspect makes a larger contribution than the shells themselves.  It is between the room and the drum tuning.  Shell quality in my opinion pays a lesser role (roll?).