Basic Stock Questions

If I put in $100 into a stock, the stock rises and I take out $120, do I only pay taxes on that extra $20?

(You don't "put in" and "take out"; you buy and sell, just like any merchant at any local market.)

Typically, yes. However, there are a myriad of exceptions (mainly, but not exclusively, to do with accounts specifically for retirement) around the world.

If I put in $100 into a stock, can I write that off as a tax expense?

No, because it's not an expense. That is because you still have that much asset (just in a different form).

What is the best way to make your gains work for you? If I put in $100, and it goes to $120, that extra $20 isnt really working to get me more money. Would I have to get out that $120 and reinvest it somewhere else, or can I make gains automatically work for me. In other words, will my stock gains grow exponentially are just linearly?

Stock prices grow or fall in a completely non-mathematical manner; they change (there's a difference!) based on market sentiment regarding:

• the competence of the company,
• good/bad luck (COVID, for example),
• general economic forecast,
• competence of the competition,
• government and legal factors,
• investor emotion (Tesla, for example),
• market manipulation (like GameStop)
• others I'm not thinking of at the moment.

An exponential growth curve can be "fitted" over two prices, but it's truly a mathematical fiction.

One thing you did not mention is dividends. This is a company distributing (hopefully) some of the company's profits back to the owners. With a Dividend Reinvestment Plan (DRIP), more shares are automatically purchased for you on the open market. An example: if you buy 100 shares now, in 10 years you might have 120 shares from dividend reinvesting. That is exponential growth.