Your only temporary options are hair extensions (which don't really work for tips of hair or bangs) or there are some spray-in colors that could cover dark hair, but they look chalky, fake and terrible.
To do this permanently, either you (or if you're too inexperienced with hair) a salon will bleach where you want the color, then dye it. Bleaching isn't as terrible as everyone goes on about, I bleach my hair every 4-6 weeks to keep it blond and as long as I use a high-protien shampoo, or specialty shampoo I'm good. There are shampoos and conditioners specifically for unnatural colored hair so def look into that before getting it done. Make sure you let the shampoo/conditioner sit on your hair for a few minutes before rinsing so you can really soak everything in.
I don't think they will strip your hair, unless you've been dying it black with drugstore box dye, in which case they can't bleach it until all that color is out.
There are no temporary ways of doing this if you want those bright colors you have to bleach your hair and bleaching your hair damages I had my ends blue and also almost all my hair purple pink. I suggest if you only want this temporary get some hair extensions in bright colors is the way to go there are also some sprays that wash out. I know Splat has some you can find those at Walgreens, Cvs etc and other places If you really wanna dye your hair go to sally's I know they have some colors and buy the bleach and the colors you want :) If you have any questions just asked I know a lot about dyeing hair :)
If you have black hair, you'll have to bleach it four or more times because it will turn orange the first time. You might even have to strip your hair (if black isn't your natural color) which is expensive. Then dye it the color you want. Seems like too much effort if you just want a style that isn't permanent. I would recommend some colorful thin extensions and try that before you make a decision. Also buy real ones not plastic ones as if you attempt to curl them they will melt. They will blend nicely into your hair.
Katys' are extensions...Ur best bet is 2 try extensions, if u've never done anything like this... & not the cheap kind, esp because you have curly hair...u may even want to go to a reputable salon if u've never tried applying any extension ever! But if your daring & sure this is what you want then bleach 2x and then put in the color. If you straighten your hair a lot, the damage from bleach is almost equivalent. You might even want to try bleaching (& color vibrant colors) just @ your tips of a few select chunks, if you like it complete the job.