Changing cassette road bike I have an 11-25T cassette with a Sora derailleur on my road bike. There are a ton of hills around me that I am struggling on when I'm on my road bike but not my mountain. I can easily remove the cassette, and I am wondering: Would upgrading to a 11-34 or 11-32 cassette be a simple swap? Would I need a new chain with more links/less links removed to accommodate the 11-32/34 cassette?)

It would be neat to have my fast setup and a climbing setup with a chain to match each cassette, though it would be simpler to use the same chain for both. Let me know if you have any experience doing a swap like this. Any information is well appreciated.
If you have the short cage version of the latest RD–R3000 Sora rear derailleur the largest (official) sprocket size is 32 teeth. The medium cage version supports up to 34 teeth.

If your current chain is shortened correctly it will probably be too short for a ≥32 teeth sprocket. Theoretically you could still use the chain and avoid the large–large gear combination but in practice that’s hard to do. Unless the chain is almost new I’d just invest the 10€ in a new 9 speed chain.

Once you have a long enough chain it should be possible to quickly swap between the 11–32 and 11–25 cassette. Some adjustment of the B-screw (the one which sets the vertical distance between derailleur roller and sprocket) might be necessary. 

Personally I’d just use the 11–32 cassette all the time. The only disadvantage are the larger gear steps and the very slightly increased weight.

See this comparison of the cassettes. If the larger steps don’t bother you, just stay with the 11–32 cassette.

I’ve recently got a power meter and it really made me realize how important easy gears are in hilly terrain. Especially if you want to do recovery rides.