Transmission?

Something going on with my 2003 Olds Silhouette. 193,000 miles. No major problems up until now that weren't caused by mechanic error.

What's going on is, when I get going after it has sit for a while - can be overnight, or even an hour, it varies - when I go to accelerate, there's nobody home. I go a little bit, it's not dead-stop. For instance, my driveway is 300 feet long. I go down the driveway (it is sloped down to the road a bit), stop at the end and wait for traffic to clear. When I get out to the 55mph road and hit the gas, I barely go. I back off the gas pedal, and try again, and eventually I do start getting up to speed. Once I get to that point, it doesn't happen again after stopping at red lights, etc. I do believe the engine revs when it's not going very much, and thus the backing-off so I don't overtax something maybe.

My first guess was transmission. I checked what I thought was the trans fluid dipstick this morning, but I think I got the wrong thing. I looked over the whole engine to find it. In the front is what I thought was the oil dipstick. In the rear is what I thought was the transmission dipstick. I pulled it out - it's really long and has this sort of fluted ball a few inches up from the tip. I'm not sure where to find it if that wasn't it.

As I am obviously not an expert, I also guessed it might be the fuel pump or fuel injectors. I have not been able to keep the tank full, been putting in $20 or so at a time and running it down to empty quite a bit due to poor finances. I wondered if this is overtaxing the fuel pump. I put $50 in today and the problem still happened.

Very much appreciate any input I can get. I can't afford repairs on this so family is talking about helping me get a different used car if this is going to be expensive.

Thanks,