I know that several have advocated using dimensional lumber (IE: 4x6s), but if you are worried about strength and capacity then look into some engineered lumber (IE: LVL, glulam, etc). They come in all sizes and lenghts (and can also be cut down from std lengths to fit your needs). When we cut the opening for our inwall tank, on the bottom floor of a 2-story house, in a load beraing wall, we used an LVL header. Multiple times stronger than dimensional lumber, and not prone to warping from the elements. Barring that go with laminating some plywood to whatever dimensional lumber, and/or your existing floor joists (as was mentioned already). The end result will be to the multiple times stronger.
Another way to maybe make your life easier is if you decide to marry some supports to your existing floor joists, rent or bottow a framing nail gun to help with the initial placement. Try use glue in between the old and the new before hand and also to carriage bolt it afterwards, for added security.
For establishing a ground support for any new joists you can always use the concrete deck blocks. The ones that are shaped like a pyramid with a square indentation in the top meant for holding a verical 4x4. Dig our a little sand, flattening the grade as much as posisble, then come behind with some gravel to shore up the ground and prevent the sand from ****ing too much underneath, then level the block and put in your support and tie it into your joist. When grading and shoring it up with gravel, do it in an area 2 to 3 times the size of what you need for the block. So if the block is 6" square, clear an area 12" from the center in all directions.
As others have prefaced their recommendations with.. I am also not a strucural engineer, and these are just my suggestions based on previous expereinces with building and supporting structures. Whatever you chose, the best of luck to ya
