well, i've been doing most of my grocery shopping by bike for about 4 years...

i use a variety of methods depending on how much i'm buying...

i have a BOB trailer for when i really have a lot and i can pretty much fit an entire shopping cart in it - i think i've had almost 100lbs in it before (i think the trailer's only 'rated' for 70lbs but it did OK) - and it's pretty stable even with 60+ lbs in it -- the BOB is awesome (also great on singletrack for camping) but expensive at about $250...

i usually just make do on the way home from work with 1 pannier and my backpack (my usual commuting gear)... i'm only buying for myself so i don't usually have soo much to buy.

but if you do a combination of 2 basic panniers and a backpack you can fit a lot for pretty cheap (i assume you already have a backpack - i have 4 now and have always had at least 2 but maybe that's just me) and you should be able to find basic panniers for like $50 or less (i have the Ortleib waterproof bags that are awesome! about $100)

i've also loaded up my 70liter trail backpack with lots of stuff, but it's not so comfortable sitting on the seat with more than 30lbs in the pack... but it works if you don't buy lots of liquid stuff to keep the weight down...

and the basket idea sounds good too... never tried it myself