When to Move
Whenever you want! You should obviously try to get moving taken care of before school starts, but some people move out way earlier to explore the area - whatever suits you and your schedule.
How to Get Your Stuff Here
Most people bring their stuff in their car or otherwise had their family drive it out when they got here. If you have a lot of furniture, etc. you may want to rent a truck or hire someone to do it (whatever works best for you). Be careful about hitching a trailer to a small vehicle - it can be more expensive than you predict (paying for a hitch, bad mileage) and it can be unsafe. Also, be aware that the speed limit is lower in California for a vehicle that is towing a trailer.
If flying out, decide what you want to take with you on the plane and what you'll ship. Seriously consider sending your packages through UPS or FedEx. If these services are too expensive for you, use the US Post Office, but make sure you get insurance! The USPS also has great shipping rates on books and dvds (media mail).
If your furniture or large items are not worth that much to you, you may consider buying new stuff when you get out here. Flying out and shipping your essentials is significantly cheaper than driving out a large amount of things (depending on how far you are traveling). Craigslist is a great place to get cheap/free furniture once you arrive in Berkeley. Some people end up saving money with this method, but it depends on how nice of stuff you already have and what you will want out here.
It is a good idea to save receipts from all moving and travel expenses, because they can be used for tax deductions and when filing for residency. Fill out a change of address application from the post office, and print/copy this for your records, as well.
For whatever reason, moving almost always costs a lot more than you think it will. Count on this.