Greyhound (or Megabus) sounds better than flying for you. Also consider hostels for the first week or month of arrival. I've used hostelworld.com with excellent results.
You can take a lot of stuff in the baggage compartment on Greyhound - I'd sell the TV, but a music keyboard (legs folded) and large duffel is common enough. Compared to flying, you get free shipping on some of your stuff, plus a lower ticket price. Ship the electronics box and remaining clothes (if any) to your initial destination.
For such a long trip, I think the key if using the bus option would be planning your food costs. In the backpack that you bring onto the bus itself, pack travel food (raisins, almonds... carrots/broccoli ...apples/oranges... bread/crackers/rolls). Refill your water bottle using the tap in the bathroom during stops. If you haven't done 2 days eating out of your backpack lately with no cooking, practice ahead of time to calibrate your actual needs, so you won't use "I'm hungry" as an excuse to buy $5 convenience store foods instead of the $1 backpack meals. The bus will likely stop only near convenience stores, not grocery stores, on a routine basis.
Check the routes and see if you have a friend en route, where you can stay over for a night. Or if there's a stop in a city you want to visit, check the cost of hostels and Airbnb; you could grocery shop at a real grocery to restock while getting some low-cost tourism in. These are more of an "extra fun for cheap" option, though, not an actual cost saver.
I have traveled on Megabus recently. It's cheap and usually excellent, but often arrives late FYI. Not sure if you can carry as much in the baggage hold as Greyhound. Check the rules for both lines because they often enforce them; for a cross country move it's worth devising a reliable plan. Maximize the savings by allocating heavy things to the bus and light ones for shipping, then winnow your belongings strategically to get the right cost saving balance after studying shipping costs.