What is the best way to take original cartoon characters that my offspring have drawn and sell them online either for the character themselves, or on merchandise as in T-shirts, hats etc.
Not clear how you would sell the "character themselves" - is that the rights behind the use of the character?
Frenchie is absolutely right that you should test whether you have a market for these cartoons first.
An even lower risk way of testing than selling at a flea market is to use a company where you can design and sell your t-shirts online. That way there is no outlay and you get a share of any sales.
You could also set up a simple website that showcases the cartoons.
Of course you would then invite everyone you know to visit. If you get some sales, look at taking the idea further.
How you would do that really depends on who the cartoons appeal to.
"What is the best way to..." depends on what goal you're trying to achieve: volume, margins, worldwide vs. local distribution, ownership of production vs. production outsourcing, licensing....
Figure out first if a) there's a market for your cartoons and then b) what you'd like your business to look like as it grows.
Personally, I'd start by focusing just on the T-shirts to test your idea and people's reactions. I'd find a manufacturer who can do a small batch of 100, and try to sell these at a local flea market, just to see if people want to pay money for these T-shirts. I'd go with the flea market option first that way I could talk to potential customers and get feedback first-hand. I'd market-test with the flea market or some other kind of direct sales channel that can connect you to people whom you have no personal relationship with (ie, no friends/family, they always say it's "awesome").
And then I'd go from there based on the results of this small experiment.