What is a good bank to use for storing my pay? Preferrably one that has free student accounts. Can I save money from my paychecks
directly to a Canadian bank Otherwise, can I connect my bank account
to my Canadian account online?
Any (almost...) bank in the US has free college checking accounts. If the bank you entered doesn't - exit, and step into the one next door which most likely will. The big names - Wells Fargo, Bank Of America, Chase, Bank of the West, Union Bank, Citi etc - all have it. Also, check your local credit union.
Do I need any ID to open a bank account? I have Canadian citizenship and a J-1 visa
Bring your passport and a student card/driving license (usually 2 ID's required).
What form of money should I take with me? Cash? Should I apply for a
debit card? Can I use my Canadian credit card for purchasing anything
in the states? (Canadian dollar is stronger than US dollar currently,
so this could be to my advantage?)
There's some fuss going on about debit cards right now. Some big banks (Bank of America, notably) decided to charge fees for using it. Check it, most of the banks are not charging fees, and as far as I know none of the credit unions are charging. So same thing - if they charge fees for debit card - step out and move on to the next one down the street.
Using debit card is pretty convenient, cash is useful for small amount and in places that don't accept cards.
If you're asking about how to move money from Canada - check with your local (Canadian) bank about the conversion rates and fees for transfers, check cashing, ATM, card swipes, etc - and see which one is best for you. When I moved large amounts of money across the border, I chose wire transfer because it was the cheapest, but for small amounts many times during the period of your stay it may be more expensive.
You can definitely use your Canadian credit/debit card in the States, you'll be charged some fee by your credit card company, and of course the conversion rate.
How much tax does I have to pay at the end of my internship? Let's assume one is earning $5,000 per month plus a one time $5,000 housing
stipend, all before taxes. Will I be taxed again by the Canadian
government?
$5K for internship? Wow... You need to talk to a tax specialist, there's probably some treaty between the US and Canada on that, and keep in mind that the State of California taxes your income as well.
What are some other tips I can use to save money in the California?
California is a very big place. If you live in SF - you'll save a lot by using the MUNI, if your internship is in LA - consider buying an old clunker if you want to go somewhere. If you're in SD - just enjoy the weather, you won't get it in Canada.
You'll probably want a "pay as you go" wireless phone plan. If your Canadian phone is unlocked GSM - you can go to any AT&T or T-Mobile store and get a pre-paid SIM for free. Otherwise, get a prepaid phone at any groceries store. It will definitely be cheaper than paying roaming charges to your Canadian provider.
You can look at my blog (I'm writing from California), I accumulated a bunch of saving tips there over the years I'm writing it.
