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,