I have a very good mountain bike and like to ride trails. I recently signed up for a mini triathalon that requires a 33 mile road ride. I don't want to buy a new bike, so I was thinking about buying a new set of rims and swapping them out for this race. Not very experienced with converting, so looking for some help on solutions. Don't want to spend a ton of cash on this either.
Even if you do not go as far as different rims, consider different tires.
Slicks (1.25") vs the regular tires (2+ inches) will make a huge difference, and usually can be found for $30 a tire or so in the US or Canada.
Beyond that, remove unnecessary hardware.
Front shocks are probably the worst thing you might have on the bike. I do not know if it is worth considering a hard fork just for this race or not.
Would absolutely go along with @geoffc's answer re tyres. I'm suspecting that's what you really meant to type when you said "rims"? But anyways...
I'd just add that it might be worth trying out some aero bars too, you can get ones that will just clamp on to your current bar. Might be worth seeing if you could borrow some first as they are very much an acquired taste, but you can imagine they'd improve your position on the bike if you can get on with them.
If possible, just rent a road bike. Even though you can make a mountain bike more comfortable for road riding, the frame itself is completely different in terms of both geometry and weight. You'll be a lot happier if, instead of spending a bunch of time and money messing with your mountain bike to make it not horrible on the pavement you instead spend about the same amount and just rent a road bike for the event.