I'm sure this is a very simple question, but I have been confused about this topic for a while now. According to Ohm's law, resistance varies directly with voltage. This means that if resistance increases voltage increases... But obviously that's not how it really works. If I add in a resistor to a circuit, the voltage decreases. I've heard that its because a resistor reduces current which in effect reduces voltage, however I don't understand how this lines up mathematically with what I said earlier.
According to Ohm's law, resistance varies directly with voltage
You should read this the other way. Voltage varies directly with current. "R" is the constant of proportionality telling how much it varies.
If I add in a resistor to a circuit, the voltage decreases.
If you have a resistor in a circuit, with a current flowing through it, there will be a voltage dropped across the resistor (as given by Ohm's law). If the resistor is in series with some other element, and they together are powered by a constant voltage source, then the voltage dropped across the resistor means there's less voltage available for the other circuit element. It doesn't mean that the voltage of the source decreased.
If you have a constant current source passing through a resistor, then, yes, increasing the value of the resistor will increase the voltage drop across it.
Ohm's Law gives the resistance as the ratio of the voltage and current, as R = V/I.
None of these are necessarily constant, all three are variables. In certain circumstances, you may be able to treat one or two of them as constant, or nearly so. But Ohm's Law always gives the ratio.
Bear in mind you can measure the voltage at a resistor's terminals with a meter. You can measure the current flowing through a resistor with a meter. But the resistance, the inherent stuff going on inside the component that gives rise to the ratio, cannot be measured independently. The resistance is defined as the ratio of terminal voltage to through current, you have to make both measurements and do a sum. That is what a multimeter does when measuring resistance.
I will emphasis on these points based on your question
"According to Ohm's law, resistance varies directly with voltage. This means that if resistance increases voltage increases"
Yes it is true ,i.e Voltage drop across a resistor increases,say if you connect a 1k and 10k resistor in series Voltage drop across 10k will be more when compared to 1k resistor.
" If I add in a resistor to a circuit, the voltage decreases."
Absolutely No,adding resistor in a circuit drops the voltage across itself but now across whole circuit i.e you may get less voltage at particular NODE but it but if you measure effective voltage it will be equal to the source(IDEAL condition)
Last but not the least
"I've heard that its because a resistor reduces current which in effect reduces voltage"
What you've heard is wrong , resistor never reduces current it just limits current i.e it slows the speed of electrons flowing in the circuit, and electric current as per wiki is An electric current is a flow of electric charge,so if you are slowing the electrons it doesn't mean you are reducing the current it simply means you are allowing limited no of electrons per second through resistor while remaining electrons are dissipated heat .
Hope this helps,