A few things to consider. Your trip is just as much a "road trip" as a backpack trip so you need to prepare for both.

1) you will be traveling public transporation so some of the light-weight gear may not hold up to harsh handling- get a large duffle to put your pack inside so it does not get wrecked.

2) you will be traveling through different climates - have your gear flexible enough to add and subtract items as weather dictates. Layering works well here. A solo pot, cup and spoon is sufficient for backpacking but you will appreciate an extra pot and fry pan for the non-backpacking parts of your trip.

3) Since you will be short on room traveling on public transporation, try to get your cook gear to nest into a small packet, use compression sacks for sleeping bag and clothing.

4) Although your backpack trips will be in relativly cool temperatures, traveling between parks you can encounter 100 degree temps. Have at least one change of hot-weather travel clothing.

5) Be aware that not all stores carry specific backpack suitable consumables- particulary butane stove cannisters. The general car-camping gas cannisters do not fit on the backpack stoves. If you use a light backpacker's butane stove, you may want to carry an extra cannister.

6) You will have to figure out how and where to store your non-backpack gear when you are doing your short backpack trips.

7) Just as important as gear is your overall trip planning. Spend the winter doing a good job of researching the routes.

8) You have an entire year to "practice" backpacking and test your gear. Just do some winter weekend trips where you live. Pick a weekend where you will have lousy weather to test out your gear.