How do I make a new wallet?
First your
choose a wallet then you install that and follow the on-screen instructions. I suggest Electrum.
What makes a signscript and witness different in a transaction, and what are they used for? What will I be using them for if I want to use bitcoin and other cryptocurrencies that call for a signscript and a witness?
You don't need to know any of that to use bitcoin, the wallet takes care of it under the hood.
If you want to gain extra knowledge; bitcoin transactions contain smart contracts that act similar to locks and keys. Each transaction contains outputs that contain the "locking script" also known as scriptpub to spend these outputs you need to provide the "unlocking script" that for legacy transactions is placed in "scriptsig" and for SegWit transactions it is placed in "witness" part of the transaction.
Execution of these two parts together will automatically determine if you can spend those coins or not.