While driving, the rpms will shoot up while the speed/acceleration responds quite slowly.
Any ideas why?
Make, model, year, mileage, previous problems?
This can be completely normal, we need more details.
Sounds like a slipping clutch.
If you have an automatic it’s inside the trans, if it’s a manual then you know about that one.
If it’s automatic then check the trans fluid, level and condition (color, smell).
2002 Chevy Trlblzr, 111k.
Also, it seems less prevelent after engine has warmed.
^
Does your Trailblazer have a manual transmission or an automatic trans?
Automatic.
^
As Circuitsmith recommended, the first step is to check the level, color, and odor of the transmission fluid. Please do that, and then report back to us with your findings.
Is this a change from previous operation? Or are you new to this vehicle? Or are you just trying to learn?
Automatics have devices inside that allow slip (the torque convertor) when you hit the gas. It allows the RPM to rise to enhance acceleration. There is also a clutch inside the convertor that locks up for cruising to eliminate that MPG robbing slip. When you hit the gas hard enough, the clutch opens, the RPM’s rise, the truck accelerates and than as you let off the gas, the clutch locks and the RPM’s drop back down. All that is normal.
Excessive RPM is not normal but much tougher to determine while driving if you didn’t note how it all works when not slipping. The burnt fluid and level check as @circuitsmith suggested is Step 1.
Since it’s an automatic (a “slush” as we used to call them), the tranny is slipping internally and needs to be looked at by a qualified shop.
With great respect to Mustangman, if it’s performing that badly it isn’t operating properly. Downshifting should be clean and almost immediate, without the engine noticeably overreving. Unless, of course, you bought my old '64 Fairlane. 
Well, given his “RPM shoots up” description, we have no idea what the OP means by"shoots up". Is it 200 rpm (torque convertor unlock) or 2000 rpm (tranny 'bout to blow)?
All respect to the same mountainbike, too, could be slipping. Not enough info to tell from here.
BTW My old GM 2 speed Powerglide was the very definition of “slushbox” IMHO. Yuck!
Point well made. Hopefully we’ll get some more details soon.
2-speed Powerglide? Wow. That goes back a few years… as far as my '64 Fairlane if not farther. 
I think all the trannys slipped in the early days of the technology. I seem to recall there was even a song about a car needing a “band job”. Can’t remember exactly.
“I think all the trannys slipped in the early days of the technology. I seem to recall there was even a song about a car needing a “band job”. Can’t remember exactly.”
“…and you know you’re gonna need another band job.” Peter, Paul and Mary, from the early 1960’s, singing about some guy getting sucked into drag racing away from a red light. But I’d have to research the internet to get exactly which album/song.
^^^ Yup, but it wasn’t a song. It was Paul Stookey do a spoken comedy routine in an album of songs. Also included bits about playing ping-pong, and ordering food at a drive-in through an unintelligible intercom as I remember.
Here’s the whole bit:
http://www.lyricsfreak.com/p/peter+paul+mary/paultalk_20292505.html
Or for the full effect:
;-]