Attempting to eliminate plastics from my lifestyle has definitely been a gradual transition. It's tough to get rid of items that we've become so used to using over the course of our lives (i.e. tupperware, plastic ziplock bags, etc.). And while there may be some items that you just can't let go of for practical reasons keep in mind that limiting your exposure is way better than doing nothing at all.
One of the ways I found to get around racing right out and immediately restocking my cabinets with all non-plastic storage items was by lining the containers with waxed paper....(this is a biggie for me at Burning Man). I also do this with my cheeses. I wrap them in waxed paper THEN wrap plastic wrap around them to keep a barrier between my food and the plastic.
If you'd prefer to take further steps to rid your kitchen of plastics then here are a few of the non-leaching storage items that I've transitioned into my home.
In place of using ziplock sandwich bags I now use these waxed paper sandwich bags with a clip to keep them closed.
For leftover storage I use glass and ceramic containers like these:
(Yes, these last two items have plastic lids but as long as your food isn't touching the lids no potential leaching will occur.)
Additionally, some of the food I buy comes in reusable glass containers with screw-on lids. These come in very handy since they're so compact and they travel well!
And one of the most recent items that I've started using (in the office, on day hikes, etc.) are these sustainable & biodegradable Disposable Bamboo Utensils. While they do specify that they're good for one-time-use I admittedly give them a quick hand washing in order to prolong their life.
I hope this helps! Oh, and keep in mind that many of these items can be found at your local Target, Crate & Barrel, Bed, Bath & Beyond and similar kitchen stores. Although Greenfeet.com is my all-time favorite one-stop-shop for all things green and non-toxic.