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Lunges, squats, box jumps, jumping rope, stretching, and actually just running will help improve speed, explosiveness, and overall athleticism. All these things can be done without weights and if done properly will help you immensely.

I agree with huff09, plyometrics is what our coaches had us do back in track. I was a high jumper and triple jumper and when me and the other jumpers started to impliment plyometrics in our training there were noticable results. If you decide to do any weight training, im a huge advocate for hang cleans to increase your verticle, they work your entire body and build your fast twitch fibers used for explosiveness. My best season has a high jumper I was cleaning the most weight and there was a direct relation to the two. If your new to lifting, though, be sure to learn the proper techniques, hang clean can be a lift you can really injure yourself in if you don't know the proper way to do it or have a knowledgable spotter.