What kind of bike do I need?

[Fishinshawn]

  I live in a small town with no real bike lanes, I live about 1.5 miles from the grocery store, about 1.25 miles from work, and fairly close to city parks and services such as the library. I have one pretty major hill between me and work, and nothing major between me and the grocery store. I want to be able to pull a trailer with two kids and some groceries, and maybe for single trips for milk or something like that a basket on the back.   I'd prefer to buy a used one, cause I'm cheap, but all I have bought before was cheap wal-mart style mountain bikes for the kids that I rode occasionally. Are there brands of bikes to stay away from?

[hyla]

I would suggest getting an older used mountain bike from a quality brand (something sold at bike shops - trek, specialized, raleigh, kona, gary fisher etc.), switching the knobby tires for slick tires.  I would add a rear rack with a folding wald basket (fits a paper grocery bag perfect).  Ideally look for a bike with a rigid front fork.  Suspension forks aren't needed if you aren't riding rough trails, and are harder to mount racks and fenders to than rigid forks.

This should fit all your requirements:
- cheap - used mountain bikes are abundantly available on craigslist, and ones suitable for a few miles of in town riding (decent mechanical condition and quality parts, but older and without all the trail specific features real mountain bikers want) can be found for less than $100 in many locales. 
- hills - mountain bikes have lots of gears, including easy ones for hills and trailer pulling
- trailering and carry groceries - sturdy frames, can handle this no problem.  To make rack mounting easier, look for bikes with eyelets on the frame that let you easily mount racks with bolts, that way you won't need to mess around with p-clamps.

As far as brands, when you see a likely looking bike on craigslist, just do a quick google search to check if the brand is a bike shop brand or a big box store brand, and only buy bike shop brands.  The only one that's a little tricky is Schwinn cause they've been sold a few times - recent schwinns (last 15 years?) are mostly big box store bikes you should avoid, while older schwinns (90s and older?) are decent bike shop bikes.     

[FreeAsADragon]

Hyla said everything, but I'll emphasize one point. Cheap Wal-Mart bikes are actually much more expensive in the long run because they keep falling apart, and eventually die way earlier than anything other brand hyla mentioned, not to mention all the energy you have to waste on them. And then they go into a landfill. So just don't get those, whatever you do.

[GuitarStv]

Your average rides are under three miles round trip.  Damned near any bike will work fine, just pick up something used that doesn't seem to be falling apart.

[WildJager]

A used quality mountain bike can be fitting for your needs as mentioned above.  Adding the slick tired and the rack and you're basically making a hybrid bike with shocks.  If you can find a hybrid on the cheap, they're basically what you get when a mountain bike and a road bike have a baby.  Lots of attachment points for mounts and thinner tires for going faster on the roads.  Usually hybrids don't have shocks since you don't need then for road or light trail riding.  However, since your distances are relatively short a mountain bike will be plenty fine.  A bit heavier and not as fast, but very prolific in the used market.

I have experience most with Trek and Specialized which I'd recommend for quality without much fuss.  And I'll echo, avoid big box store bikes.  They are just awful.  I bought one for a temporary job gig far from home just for a cheap ride, and the chain broke after only a few days.  It was a mountain bike.  I mean, how do you even break a brand new chain from normal riding?  Lesson learned.