First off, welcome! And congrats on taking the plunge to get feedback :)
For more detailed advice, I'd recommend taking a look at the Case Studies board and the
How to Write a Case Study thread.
Some preliminary suggestions based on the budget you've posted:
Heat - This looks very high to me, what is the climate where you live? And does this cover your renter's heat or just your own portion? You might be able to get an energy audit from your local utility company, which might suggest some fixes to bring this down. Leaking window/door frames are a big one, or you might be able to improve insulation in certain areas.
Cable/Internet - Do you really need cable? Or can you persuade the family to manage with Netflix/Hulu/Amazon Video, which I see is already in your budget? Shop around for cheaper internet options, depending on your location you should be able to do better than this.
Gas - This is a lot, but should go down once your commute is shorter
Grocery - This isn't too bad for a family of 4 (btw, congrats on the soon-to-be 5th!), but could possibly be cut further by careful shopping if your wife is on board.
Cash - Figure out what this money is actually going toward. Clothes? Toys for the kids? Travel? Hard to know if it's too much without knowing what it's being spent on.
Gym - How often do you go to the gym? Can you replace this with a set of hand weights and runs outdoors?
Car Payment - This is a big one. What kind of car, how much do you owe, and what is it worth now? Can you sell it and buy an older/smaller car to replace it? Preferably one with better gas mileage, too ;)
Best of luck!