I've tried to move stuff in a variety of ways with a bike:
- Backpack . . . OK for short, light trips. You already have one most likely, and your backpack can get double duty when you're not biking. It sucks because it makes your back sweat, and it puts additional weight on your wrists/ass while biking around with it loaded. I don't find a backpack makes any real difference to bike handling for me.
- Rack w/Milk crate . . . ideal for cheap light dump n go storage. I liked my milk crate, and it didn't make riding the bike significantly harder - EXCEPT AT LOW SPEEDS. Carrying 20-30 lbs of groceries was a bit scary starting/stopping, and the crate seemed to catch more crosswind in the winter than anything else I've used. The milk crate does look kinda ghetto, and usually isn't easy to detach so will cause problems trying to use those bus mounted bike carrying things if that's a concern. I did a milk crate lower and on each side of my rack for a while, but it makes your bike annoyingly wide and catches a shocking amount of wind when trying to go quickly. Cars give a wider berth when passing though.
- Rack w/Panniers . . . ideal for riding to work. No weight on your body, usually easy to remove from the bike when you want to go in, relatively inexpensive if you shop around. It's easy to find panniers that will fit your clothes and a lunch (and bike repair kit). Awkward for groceries as these bags tend to have funny shapes, and panniers take more time to remove/attach than just dropping stuff in a basket if you're in a hurry.
- Rack with folding baskets . . . great for groceries or any shorter trip! These things are the bomb for groceries . . . you can get one large bag of groceries in each, and they're pretty sturdy so you can load a bag of potatoes over the top of both. Steering is minimally effected because the weight is lower. You get on and off the bike in record time, just need to grab the bags in the baskets. They fold up for easier bike storage. My main issue with these is that the baskets tend to be pretty heavy . . . so doing long rides becomes less fun. It's not easy to remove these if you want to do a fun ride on the weekend either. They're tough as fuck. Mine took a hard crash on ice in the winter, sparing my leg much of the road rash . . . and after banging them out with a hammer they worked fine.
- Trailer . . . big storage! Lots of room, can carry very heavy things, doesn't really make much difference to steering. It's expensive, it's bulky (can be a PITA parking / requires a different locking up strategy), it weighs more than any other option (but you can detach it when you're not using it). An added bonus is that people think you have a kid, so often give you more room on the road.