Just to open with, I have done some research on sprites and seen some other posts on this forum. I am relatively new to programming, and lately I've been making games in which complex characters have to move, but they do so slowly. I need to make a custom sprite so that it moves quickly without slowing the game down. Text sprites, plot sprites, and those odd overlapping text sprites don't seem to be what I'm looking for. So I have 2 questions:

1. How do I make a custom sprite (one that I can design pixel by pixel)?

2. Is there a way to do it in plain basic, no assembly?

I am using a TI 84 Plus CE

I know, in plain basic, a color sprite that's not text is going to be significantly slower than an assembly or C program. I'd suggest learning C and using the toolchain made by MateoConLechuga, because Sprites will be a lot faster there.
In Basic, you'll have to do everything pixel by pixel but you can use lists, matrices, and algorithms to make it a little faster. See tibasicdev for some optimizing techniques.

Although it is possible to work something out, it would be quite messy. As stated above, if you are looking for efficiency and speed, you will have to look at C. As for text sprites, they will not work on the ti-84 plus CE because of the way characters are displayed (with an extra white border) so unless the sprite is 5 pixels tall or less, it will never work.

If you program in TI-BASIC, you can use Sprites v3.3 by grosged: https://tiplanet.org/forum/archives_voir.php?id=539203

It's split into multiple sub-programs and lacks any error checking (meaning that you must backup every few minute because if you do any mistake you will get a RAM Clear), but it supports sprite scaling, rectangles and has a text routine.

You may find this helpful:
http://tibasicdev.wikidot.com/graphics
It contains information on several different types of sprites.
Also:
http://tibasicdev.wikidot.com/textsprites For more specifics on text sprites