I have no experience in this field (single with no kids) so my opinion might not be all that valid.
Personally, I'd like to stay home with my kids if I had the choice. Kids are amazing and I'm sure it's even more amazing when they are yours. My mom has told me tons of stories about how when I was <insert age> I did <insert something>. It's sorta embarrassing to hear these stories all the time, but then you realize that this person truly cares about you enough to remember this story. I'd like to give that to my kids instead of having a sitter or someone else have these experiences. In addition, what happens if the kid is at school and gets picked on? I'd rather be there to greet them after school and talk to them about it then hear it hours later if at all. My one friend came to my house and started crying about something that happened to her at school and explained how she told her parents when they got home from work but they (sorta) told her to get over it. I'm not saying you'd be like them, but if a kid is hurt over something, it's better to be there for them immediately then when work gets out.
Now, as far as the finances go, I'd just like to point out that you do have a job. It's not like you are just sitting at home not doing anything all day. Generally moms do tons of work throughout the day and on top of that, you are going to a night job. Obviously it's not enough in his opinion, but you aren't just dumping all the bills on him. Paying for all that is going to have to factor in and if $9 is your starting salary, you'd almost be better off staying with your current situation and pulling a few extra shifts.
I'm kinda with vickysue on how it sounds like he wants his life to remain the same but hopes you'll change to make it easier. If he wants to go back to school, he could take 1 or 2 courses at a time and still be fine. It's not like he has to go back full time or anything. Plus, does he think if you get a full time job that he can just quit and focus on school? It doesn't sound like that would be possible even if you get a full time job so why have your kids lives, your life, and his life all change at the same time. That just seems like a lot of stress to throw on a new marriage.
One question I don't think anyone has asked yet is do you guys really need the money or are you stable at the moment and he just wants to do a bit better? You don't have to answer that, but it's something to think about. There nothing wrong with wanting things to be better, but he should be the one trying to change first before saying you should.
I'd also like to point out the fact (at least when you posted this) that he's your FIANCE. You aren't even married yet and he thinks things should change. That kinda screams 'what the heck' to me but obviously I don't know everything about the situation. If this is a soon to be new marriage, kinda worries me. Again, I've never been married, so perhaps I'm wrong or I'm taking this whole thing out of context, but it just sorta bugs me when others think they can tell you what to do and you should do it. I mean, heck, even if you both sit down and figure out that you working full time would be better, it's still up to you to do it.
As others have mentioned, you both REALLY need to talk. In fact, you might even want to set a date with him on when you both can talk about all of this and kinda warn him you'll be researching these things so you both can figure this out. That then gives him some time to do research as well to sorta support his side so it isn't like you're ambushing him with research to support yours. You should also look at your bills currently and see if there is anywhere you could cut back to help save. For instance, I don't know what you're feeding your kids, but when I was a baby, my mom said she bought actually vegetables, blended them up, poured it into ice cube trays and froze it, and fed us that instead of the jarred baby food. And by using the ice cube trays, you can adjust the amount of food very easily.
I wish you the best with this situation and I hope you find the best outcome.