First off, educate yourself on jumpseat procedures and etiquette. For security reasons I won't go into too many details here but the info should be available through your employer/union.
I would plan on a few nights in a hotel at your new domicile, you probably don't want to commit to a crashpad sight unseen (unless it's recommended by someone you know). Crashpads can be found online and through crew-room bulletin boards. Be aware that in some cities they are illegal (too many occupants)...actually they're probably technically illegal everywhere but only a few places try to enforce it so they have to fly under the radar. In that case you may have to ask around.
When selecting a crashpad, make sure you have suitable transportation to/from the airport. I pretty much always brought a car, but some places like ORD and NYC would be hard to park, involve permits, and already have good public transportation.
If you'll be spending a lot of time at the crashpad, make sure there's stuff to do nearby (gyms, food, shopping, etc) or again have transportation.
Some crashpads do not allow reserves, only commuting line holders who spend the night and then leave. If you're on reserve make sure the crashbad is suitable for that.
The 22" stealth suitcase fits in most overhead bins, and can hold 4-6 days of stuff. You'll probably need to stash additional clothes at your crashpad...they usually have a cabinet or bin for each resident.
As for food...flying hungry is dangerous, kind of like flying tired. If you can't afford hot meals make sure you're getting a combination of carbs/sugar/protein...PBJ's are a cheap solution for that. Ideally your crashpad will have a kitchen, rather than just a hotplate/microwave. Some airports have good food (ex SFO), others don't but it's all expensive. If you carry perishable food in a soft-sided cooler (available at online crew stores) you can ask for hotel rooms with fridges...sometimes they'll bring a small fridge to your room if you ask, or let you put your food in the hotel fridge. You can also leave it in the crew-room fridge if you're doing hub turns.