Please do not use box dye.
This goes back to the first "Color does not lift color" rule. Nothing will happen if you put pink dye on brown hair, as I learned in the seventh grade. You'll have to lift it to blonde before the dye will have an effect, unless your hair is already super blonde. There are more products out now to tint hair that isn't extremely blonde, but only in shades darker than your natural color. They will provide a tint but not a clear color, and only if they're darker than your hair.
You're going to have to tone it to an ashy or neutral color first, or add some of the opposing color. For example, you might want to put a purple-blue on your hair to counteract the yellow tones. If the color is darker than your hair (a dark blue or purple, for example), it may cancel out these tones a little better, and even give a bit of a tint on lighter-brown hair (for example, if you use Overtone for brown hair). However, these colors are still mixing with the brown of your hair and will not be super bright. If you want bright blue, you'll have to go lighter.
Color dye does not use developer; it only deposits color (the good news is, that means it's less damaging). It must be darker than your base color, meaning that your base color needs to be super light. I'd highly recommend going to a professional for pastel colors so they can make sure you lighten enough and get the proper tone.
It's sort of a double-edged sword: Pastel and colored hair fades really quickly, but it can feel almost impossible to eradicate. You can do something like a bleach wash (where you mix the bleach as usual, then add an equal amount of shampoo and lather, keeping it in for 20–30 minutes) to fade remaining dye, but it's usually not gone completely. It seems to cling to your bleached hair. That being said, you can usually put another color over it and be fine, depending on how faded it is and how dark your new color is. Don't forget what I said about the color wheel, though! If you put red dye on blue hair, it'll come out purple!
When you're lightening hair, heat can be useful. You can use a blow-dryer to heat your hair as it's processing to make it go faster or be more extreme. However, I would not recommend doing this until you know what your hair can handle and unless you've been dyeing your hair a long time, since it can also damage your hair more.
I highly recommend watching YouTube videos first and making sure you have all the correct products, including foil. One method you might find harder to mess up is going for a balayage look, where you decide exactly where you want each highlight to be and use plastic wrap between layers. You don't end up with zebra stripes, but make sure you use a super-thick cream bleach so that you can see exactly where you're placing the highlights. You'll need a steady hand and a really good idea of where you want each highlight to be.
If you especially hate the stripes at your roots, you can use a slightly darker color there.
If you're just adjusting a few shades, it's a quick process! But if you're really changing hair colors, be prepared for it to take a while to achieve the exact look you want. The most important part is giving your hair some much needed TLC. When in doubt, consult a professional and watch YouTube videos by professionals. It's especially hard to use dyes on your own head, so don't go with anything extreme your first time.
Obviously if you're doing a drastic change, you need to keep in mind the health of your hair. But I hear so often that a bright color is "clownish" or that you can't have hair lighter than your skin tone, which is absolutely ridiculous and based on antiquated Western ideas of femininity and beauty. Do whatever makes you feel happy and confident, and remember that at the end of the day, it's just hair!