No that's not normal. It could be many problems. It sounds potentially like a clogged valve body. The valve body controls oil pressure which is used to switch gears and if it gets clogged you can have shifting problems.
If it is clogged, you need to take it out and pressure wash it out, and may have to switch some seals and plastic check balls. It is a complicated issue and we don't have the right competence for it here. It would be better to get a good used vb and sub it in and see.
First, check if all your solenoids are within correct resistances. Use the TCM's harness and check resistances through that. Download the ETM for your transmission (should be 5hp18) and check using a digital multimeter.
Then check to see if oil levels are correct. The bentley manual will show you how to check this, there's a unique method involved.
Then tell us if this behaviour is there only when the car is cold, or when it is hot as well. That could be significant - would basically indicate insufficient oil, you'll have to topup and check for leaks.
You could have a clutch problem.
2 seconds to initiate shift is waaay too long.