It's OK. Your three-year-old is not crazy, and you are a great mom. This is a big transition time for everyone and you'll get through it.
Here are some things that have helped me with my three sons . . . The book "How to Talk so Kids Will Listen and Listen so Kids will Talk" is beyond excellent. No mental gymnastics, just simple communication strategies that really work. I cannot recommend it highly enough. This book has been around for awhile and you could easily find it at the library or cheap online. The things I learned from that book have helped me diffuse so many tantrums and conflicts. For example, it's really reassuring to a child to validate and name their feelings, even if you are correcting their actions. So you can say, "Wow, you are so frustrated! It is not OK to hit," or wish-grant: "I understand you really wanted that cookie. If cookies were good for you, I'd let you eat them all day," or provide another outlet such as "You are disappointed we can't go right this minute! Can you draw a picture/stomp your feet to show me how disappointed you are?"
It's also important to make expectations clear in advance: "We are going into the store. I need you to be a gentleman. That means I need you to keep one hand on the cart and walk next to me and help me put things in our cart. If you run or grab things, you'll have to sit in the cart. Let's have fun and you can help me choose fruit and ice cream."
I also learned that children this age need positive instructions, so avoid using "don't," "no" and "stop," very much. It's just too abstract. If you say "No hitting! Stop hitting! Don't hit!" he hears "hitting! hitting! hit!" Try clear directions such as "I need you pet the kitty gently."
Giving instructions as they relate to you is also important. Starting requests with "I need you to," "I want you to," "I expect you to," or "Let's" has saved me from started every direction with "Aaaagghhhh! You're killing me!! WHY can't you LISTEN!!" But that probably doesn't happen at your house. ;) And if it does you know there's hope.
My son's preschool teacher, who has a masters' in child development and 25 years of experience, says children misbehave because they are hungry, tired or legitimately need positive attention. Keeping him well-fed, rested and his "love bucket" filled will help keep things from getting awful too quickly. I know lots of people will disagree with me, but I rarely use time-out, and when I do I pitch it as a welcome break instead of a punishment: "You are really angry with your brother. Go have some time out in your room and I won't let him bother you in there for a few minutes so you can get back in control of your body." Everyone needs to cool off sometimes, I know--me included-- but if you do it as a vindictive punishment or very often, it just gives them an opportunity to sit and sulk and seethe about injustice and plot revenge. . .that's hardly the intended purpose!
If you've got a new baby, it will be terribly hard to enforce a time out right now, too. Consider offering lots of "time in," where you read or do stickers or make English muffin pizzas or do puzzles when you see trouble brewing. I kept a little basket of small fun treats next to my nursing chair and would take them out when I was nursing the baby so my older child would not be completely ignored and start to misbehave.
La Leche League also saved me--they publish a great book called "Adventures in Gentle Discipline." It has lots of great ideas. Going to their meetings helped me learn about how to enjoy nursing my baby while maintaining my loving connection with my older child(ren). There are a couple groups in the Sandy area and I think you'd like to meet the moms there. You can find a meeting at www.llli.org.
Hang in there, A.! Best wishes!