How to start moving on Battery power?

Yes, this is normal. The Prius is a HYBRID vehicle, not an EV. With a fully charged (8 bars green) battery, you MIGHT be able to go from zero to 41 mph. BUT it would probably take 3 mins to accelerate to that speed and you would have to be on level ground, no turns.

You really DON'T want to be doing this. Not only because it takes so long to get to speed, but because (unless you've had a PHEV conversion) you are wasting gas. I know this sounds illogical, but you need to realize that the only way a non-PHEV Prius gets electricity into the battery is by running the gas engine. Conversion inefficiencies (gas to MG to electric to chemical [battery], back to electric, to MG, to wheels) kill any mpg savings you think you might be getting.

 

The engine has to be warmed up to run just on battery power.

 

An 06 here, installed the switch that allows the prius to move around without the engine starting within the first minute, if I start with a full battery I can go a few miles before it depletes the battery, it's good for moving the car around my residence without alerting the dogs, so it's a stealth thing. Other wise after the engine is warmed up, just pressing on the potentiometer slowly, will allow the car to run on electric until the battery decides it's had enough and makes the ice engage.

If your battery never gets the chance to be your only source of motion I would check the condition of the 12 volt battery, it may be telling the computer it's never full so the ice is continually trying to charge it...

 

As stated by Ursle, you can install EV mode button as a mod.
This button was not included by Toyota on USA gen-II models, but may have been included in other countries. Probably a lot of older discussions on Prius Chat for this mod.

 

The electric motor has 295 foot-pounds of torque, accelerating to speed on electric only is NOT a problem. What it doesn't have is the horsepower to maintain speed, and that limitation is the battery, not the motor. A plug-in with the same engine and motors will happily accelerate to, and cruise at 65 MPH.

Only being able to go 3 MPH in electric mode, once the engine is warmed up is NOT normal. I have no trouble going 25 or 30 in electric mode in my 2007.

For the most part, the car does a good job of choosing when to be in electric mode, and second guessing it requires some significant knowledge (both in terms of how the car works, and what is soon going to be asked of it).