My mother never really taught me anything about credit or finances. The funny thing is I would see her with a ton of credit cards and she used them frequently. What I didn't know is that she always PIF and never carried a balance. In addition she made extra house and car payments each month. She had great credit, and paid very little interest to anyone. Now she is married to my step dad who is a very conservative spender but was taught cash only. When they married he still had his student Visa account and no other credit cards. By this time he had been running the ranch for a decade, but was paying higher interest rates on land loans and opperating loans as he did not know how to work credit. My mother changed this and had all of the loans paid off 5 years early. Now they have added more land and have great terms on the loans they have for the business.
Now that I have been reading on here and researching credit I have talked with my mom about what I have learned. She was well aware of most of it, and even had many suggestions for how to moniter my spending and such. I was amazed what she knew and never told me about, her response was that she figured I would learn it eventually. So while I am following her now, I am also teaching my kids about finances and credit. They have AU cards on my accounts for things like lunch when we are skiing (they ski with a team so I am not around when they eat) and they have to report back to me what they spend (preferably with receipts but still working on that reminder). I have explained in general how credit works and how budgets are setup.
Part of my push to teach my kids comes from their mothers spending habits, she will spend until all accounts are maxed, make minimum payments and then spend that. One of my boys came to me one night very scared and told me he did not want to go skiing anymore. After some tears I asked what was wrong, he had a hard week and was getting in trouble for not doing chores. Well it turns out he was afraid we were "almost out of money" and did not want to do chores so I would not have to pay him. I realized he had no understanding of working and getting paid and his concept was there were "accounts" full of money and when they were empty you were done. That came from his mom telling him they were broke and could not afford groceries. So I sat both my boys down and explained working, budgeting, credit and savings and how they allowed us to continue to enjoy life as we were. They are only 10 so I gave them overviews, but it helps and I continue to include them in what I do with the budget.
So I will say this, kids see what is going on and they pick up on small comments and especially the stress of finances. I always thought I would just shield them from the complicated matters and it would be easier on them. It was actually the opposite, so now they see what we spend on, how the budget is figured and a general sense of at least my household finances. Their stress has gone down massively at me house and they seem to enjoy learning about this stuff. So even if you cannot give them an allowance, maybe explain the overall budget and how you are working to better the household finances. Include them on long term plans, my boys are great at reminding me that we are paying for the boat by not eating out as much. If we go out twice in a week they will skip the next week, although they have both said they prefer to eat at home anyway 
so it is normally my not wanting to cook that prompts us going out. Another thing they have been stellar at is turning of lights and electronics, I showed them the utility bill and explained how to save energy and reduce the bill. It changed from me chasing them around the house turing off things to me telling them to turn on a light when they read.
My parents are a dichotomy when it comes to finance. They divorced when I was about 11. Not too long after that my father was transferred to the Middle East for his job. This meant that I had to live with my mother. My mother cannot spend within her means. She THINKS she does, but she buys many unnecessary things. She was a very likeable, but very absent mother. Eventually, my dad moved back to the US, and I went to live with him. He is wonderful with finances, and he got me a checking account. But that was the end of that. I never bounced checks or anything, but I got involved in credit, and my mother's upbringing was very, very strong in my mindset. I made many mistakes. Eventually, in my early 20's I had to declare bankruptcy, because I was making next to nothing and had many debts. My actual amount owed was really not ridiculous, since the limits were so low......maybe $3000. My father, rather than bail me out, paid the bankruptcy charge. THIS IS THE THING THAT I THANK HIM FOR THE MOST. If he had paid any/of my debts, I would never have grown, and I would have ended up in ten times the trouble. It was an amazingly huge deal that I did not realize until later. 10 years is a long time. And it was so smart on his part. My BK will fall off in October, and I guarantee you that I am better for it. I have had to deal with 10 years of learning how to live within my means. I only got back into credit in September of 2013, and I am so strict with it all. He could have easily paid it, but he did not, and in retrospect, I honestly love and respect him for what he did and saw in my future at that point in time. And, now, I have actually tried to start teaching my mother about credit. I am cautiously optimistic, but it will take work. I know she wants it, but I want it to be very, very, very slow. I worry that she will revert to old mindsets. Best of luck with your children. It is okay to teach the hard financial lessons when required. They will thank you for it, but it may take many years. 
I do not recall a single conversation with my parents about finances. I know they both handled their money well since we never had any issues with money and that has carried over into their retirement where they live frugally but still enjoy themselves. For some reason even today if I ask a financial question I get very short, curt answers. Its like i'm a reporter asking Bill Belichick a question.
My son (8) gets a weekly allowance that is seperate from any chores. The catch is that that allowance is his to spend anyway he pleases but that he alone is reponsible for purchases. If he wants a new toy he must save for it. If he wants to see a movie he pays his own way. My goal with this allowance is not to penalize him but rather hopefully to show him how quickly money will dissapper without a plan.
Starting Score: 580