I will echo what most others have said as far as furniture. Once you have sold/donated/otherwise gotten rid of furniture, take a look at your books and DVDs. You could donate/sell anything you don't need, the rest can go via Media Mail for fairly cheap which will help lower the total weight and volume of anything else you're shipping.
If Amtrak serves somewhere near your origin city and new home with stations large enough to accept cargo, you can ship most stuff (no art, electronics, glass items) at a rate of about $90 for the first 100lbs and ~$.75 per each additional pound. Greyhound also has a similar service for pretty cheap, but the routing can be restrictive.
If you are flying, depending on your airline you can get a good chunk of moving capacity that way, too. The maximum size is usually 62 linear inches with a max weight of 50lbs before being charged for heavy bags. If you don't have or can't find free boxes, U Haul's "large" size moving boxes are 18"x18"x24" which comes to 60 linear inches. Most U Haul stores have a take a box/leave a box bin where you might be able to score a couple for free. If you end up paying they're under $2 a pop and you can return unused boxes to U Haul for a full refund. (These boxes work well for Amtrak, too.)
Southwest would give you two checked items of up to 50lbs for free, and even if you were to fly with an airline that doesn't give free baggage (United for example) it would only cost $60 to check two boxes of up to 50 lbs each.
Some links for you:
Good luck!