Hi Kristi,
I have a little boy who just turned 5, but at 3 he was really pushing the buttons too. My advice is BE CONSISTANT... if after he has been in time out, he does the same offense that sent him to time out in the first place then maybe do the same: send him back to time out for as many minutes as his age(if he is three, send him to time out for 3 minutes). Keep up this pattern until he is tired of being sent back to time out...
if you give up, he will know that he can wear you down and you just want to prove that the behavior is unacceptable.
Another trick that works for me is that if your son has a favorite toy, tell him that if he does the offense again, that he will lose his favorite toy for 2 or 3 days (whatever you think is fair) and then put it up high (like on top of refrigerator (or whatever) --- somewhere that he can see it, but not reach it and then keep it there for the amount of time you said you would. don't back down. This has worked for me too. (and most likely this behavior is your sons way of trying to get your attention since your attention has now been shifted to a new baby...maybe, try to include him with helping with the baby as much as a three year old can...
("Can you help me change the baby's diaper? Get the wipes and the diaper please..." ... "What a good helper and great big brother you are... " "the baby really loves you")
When he whines, don't give him what he wants JUST BECAUSE he's whining.... or you will be "rewarding" that whiny behavior.
let him know that NO means NO, and stick to it.
Just be consistant and do what you say, follow through.
Hope this helps, it has for me...