I'm pathologically “green”.
Not in the Kermit-the-Frog sense, of course, but in the ecological sense. Paper, not plastic, thank you. Better still, I bring my own bags — I have a string bag that expands to hold surprising amounts of groceries and amazes the cashiers at the store; I have a canvas bag for times when I need something sturdier. I keep the string bag in my car, so it's always there.
That's part of the first element of the Holy Green Trinity: Reduce, Reuse, Recycle. If I don't take bags, if you don't take bags, if you and you and you don't take bags, “they” will eventually need to make fewer bags. If we buy products with less packaging there will eventually be less packaging produced. Reduce.
If you don't have a string bag or a canvas bag, you can always put some of the store's bags in your car and take them back with you to use them again. Some stores will even give you a couple of cents back, as an incentive to do that, though that doesn't motivate me much. A few cents per grocery order is nothing; thinking that I'm doing something infinitesimally small to help keep the waste stream under control, well, that seems more like something.
I keep plastic containers from take-out orders and use them to store leftovers, and I never buy disposable containers at the store. I collect what bags I do get (and sometimes take “donations” from less-green friends, who have enough bags to fill the Grand Canyon) and use them to hold kitchen garbage, so I never buy garbage bags. Reuse.
I keep track of what I can recycle in my area, and I recycle everything I can. I can get rid of plastics numbered 1 and 2, of newspapers, magazines, corrugated cardboard (alas, not paperboard, as from cereal boxes), junk mail, glass jars, and cans. It's convenient: the town picks it all up at the curb... but I'd take it someplace if that were what I had to do.
I compost, which is natural recycling. Food scraps want to rot; it's what they do. You can read all about making the ideal compost pile, with the layers and the mixing this and that and the turning and the.... Skip that, if you don't care to spend the time. You don't need to buy a special contraption, either. Just find a place that's out of the way and pile your fruit and vegetable scraps there (not meat, because that'll attract scavengers and other pests). Put the tea bags and the coffee filters there too. Pits, peels, whatever, any time of year. It goes away, all by itself, and you'll be amazed at how much more time it takes for your garbage bin to fill when you're not filling it with all that stuff. Recycle.
Make sensible choices, as they make sense for you. There are good reasons to use disposable items; use them when they make sense. But when it makes sense not to use disposables, stick with the real things. I use a ceramic mug in the office, and never take a disposable cup. I bring a plastic travel cup with me, when I drive or fly, and I avoid using disposable cups. It's easy to pack. I use cloth napkins and I avoid using disposable plates and flatware. When there's a good reason to use paper plates, though, I do. The key is that I think about it and make the choice.
If you have kids, make a game of it. Have the kids think of ways to avoid using disposable items, and have them learn when they should use them. Let them manage the compost pile, and show them the nice, rich, black compost they get in the end. See if they can find fun ways to reuse things that you'd otherwise throw away (but strike a balance, don't just save things and build up clutter — and those people who figure out things to do with dryer lint are just taking it too far). Get them to blog about it, to share their ideas with their friends.
Spread the word; the word is “green”.
 Apart from the minimal discount they give, 2 cents per bag at my local store, I'm always amused that the formula they use is based on how many bags of my own I actually use, rather than on how many of their bags I don't use. I stop at the store on the way home frequently, and buy fresh items for the next two or three days. So I almost never have more than the one string-bag-ful of groceries — remember, it expands. Had they bagged my food in their plastic bags, they'd often give me three or four. But they still count it as one. So it's a good job it's not that that I care about.