RPMs shoot up

I have a 94 Honda Accord. I got it from a family member, and ever since she got it, the transmission has always been a little hard when it shifts. Lately, the engine will rev up when I get above 30 mph. I constantly have to take my foot off the gas pedal to make the rpm’s go down as the car accelerates. To accelerate I have to do it slowly. My water pump has a leak, so when I get that replaced the timing belt will also be replaced. I’m not sure what the problem could be.

Sounds like a stick with a bad clutch. Get if fixed right away or you will not be able to drive it at all.
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If this is an automatic, you might want to take the thing to a transmission shop before you pay for the time-belt service-- if your transmission is on the way out, your car is probably at a point in which you need to do some serious pondering as to whether you want to keep the thing going.

Does it have automatic transmission fluid? On Honda, you check the fluid level with the engine OFF.
Is the automatic transmission lever hard to shift? If so, the transmission may not be set properly in each position.
The automatic transmission sounds like it beginning to slip. So, a visit to a transmission shop is in order.

“So, a visit to a transmission shop is in order”

And I suggest a independent local transmission shop. The chains tend to end up costing you a lot more for work done by the high school kid the hired last month and the manager was selling used cars last week.

Thanks for your help. I just wanted to add some more information in case it could be another problem. The transmission is automatic. If I start the car and wait, the RPMs go up and then when I shift into Reverse or Drive the car seems to not have the power to move quickly. It eventually gains full power when it shifts the first time.