For my project, I want to read analog sensor inputs at a very fast rate, at least 20kHz. What would I need to do? I read that analogRead() is too slow, but I'm not sure what else I could use. I'm currently using PuTTY to capture and save the serial outputs. Would I have to increase the baud rate as well?
Maybe.
Fast sampling ADC - Using Arduino / Project Guidance - Arduino Forum
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You forgot to mention which Arduino you have.
20 kHz analog input data rate is no problem, and the upper end Arduinos (not AVR-based ones like the Uno R3, Mega, etc.) have default analog input rates much higher than that.
20kHz @ 16bits/reading (assuming no bit packing) + overhead of 1 start, 1 stop bit 20,000 * 18 = 360, 000. So yeah, you'll need to bump up the baud rate.
You can get some data compression (and thus use a slower baud rate) by taking advantage of the fact that the ADC readings are 10 bits each. So you can fit 8 samples into 10 bytes instead of 16 by packing them together. That's left as an exercise for the user 
I'm using the uno. So in essence I could still use analogRead but with a higher baud rate, or would I have to use ADC with a code similar to this https://www.gammon.com.au/adc