The toilet bowl is filling up very slowly. When I take the back of the toilet off, I see that there is only a little bit of water going into the fill up tube. I then found that if I turn off the water to the toilet and then turn it back on again, the water pressure is perfect and the toilet fills up normally. Do I need to change the water shut-off valve or is it something in the tank that needs to be fixed?
Some fill valves turn the water down if they detect a leak, sounds like yours is malfunctioning, that or scale, rust, gravel sticking in the valve. Some can be cleaned by turning the water off removing the top of the valve and turning the water on just a little. Putting a cup on top of the valve allows a higher water pressure. If the fill doesn't improve a replacement will be needed. Not all fill valves can be cleaned.
It could either be the shutoff valve or the fill valve in the toilet. If your shutoff is a gate valve (turn it many times to open and close), these sometimes are affected by mineral deposits. Replacing this with a quarter turn ball valve is pretty straight forward and will ensure reliable service for the indefinite future.
Replacing the fill valve is also pretty straight forward and these do indeed fail.
I would probably start with the shutoff and if that doesn't solve it move on to the fill valve.
I had the exact same problem. For me trying to clean the fill valve of sentiment did not help nor did the cup on top.
For me, I needed to replace the fill valve seal which has gotten a bit distorted over time. I went out, and for less than $3, bought a Fluidmaster 242. Just slide off the old seal and slide on the new seal after removing the top of the fill valve.