raise sunken floor
There are some issues here that you should solve before even adding any lumber.
1. moisture/venting-you are creating kind of a crawl space under the new floor. Although not as bad as earth crawl space, you should plan on being able to dry out that space. This is more important if you were thinking to add insulation.
2. ceiling-I assume it is high enough, but normal height is 8 feet and some light fixtures and most ceiling fans need 8 feet.
3. windows-there is a rule about distance between floor and window, especially if drop off to outside is high enough to be dangerous to kids, for example. Sorry, but I can't remember the exact rules.
4. electrical outlets, switches-these need to be certain height from the floor, especially the switches.
5. heat baseboards/registers-if water heated, will require plumbing skills and some planning on how the water will flow with new piping. If hot air, maybe new ducts.
6. what is the original floor?-concrete, lumber? If concrete, does it stay dry or has moisture issues? If lumber, dimensions and span so that it can support the weight of new floor.
7. New floor-is not a structural part of the house, so you may be able to use I-joists or just use plywood to raise the floor. This will cut down on the weight. You can also level the floor if that is also an issue, but you have to keep the other floor in mind since you want to match two floors.
8. Finish trim may need new blocking behind the sheetrock, although you can use glue to the sheetrock, but this is weak.
Draw up a plan with the finished floor and work down. If you want a seamless transition, your work must be exact to 1/16" or less, but you may have to refinish the other floor, if wood floor. First you should have a water level at least, good quality 4 foot level, 6 feet or better straight edge and thin line chalk line. Average chalk line does not give you line precise enough. As a beginner, I am assuming, always wear construction grade gloves to protect your fingers.
Have the finish materials, flooring especially, at hand when making the plan. Make sure the flooring will be available when you are ready, not something that may be discontinued or sold out.