I want to take down the ceiling above my kitchen to expose the roof above, but I need to know if I should leave some of the joists in place for support of the side walls they are laying on.
If you remove any joists it will change the structural load. You need to have a Civil Engineer design a new structure that can replace the joists (if that's even possible).
We'd need a lot more information. Is your roof a joist+rafter roof or a truss roof? Where is the insulation? How much insulation do you need? What are you planning on finishing the underside as?
You definitely don't want to be removing structural elements without talking to an engineer. You can still raise the ceiling and retain your joists and/or trusses, though.
If you have a truss roof, you really can't remove any part of that. You'd have to rebuild the roof entirely. If it's a joist + rafter roof, you might be able to remove some of the joists--or even all of them, provided you replace them with something to keep the walls tied together...such a beams or cables.
If you want to remove all structural members, then you need to look at getting a ridge beam. In that situation your roof is supported by a giant beam that runs the length of your roof's ridge, held up at both ends with the appropriate wall/post structure.