Congratulations for getting the kids involved. Yes you should definitely start from seeds, that way the kid could learn about science at the same time. You can start tomatoes and flowers seeds indoor now.
All the vegetables you listed are great for beginner but donÂt forget to add some sun flowers for the birds and marigold to keep the bugs away and bring in the bees. The cheapest way to buy seeds is at Walmart, some started at .20 or at Paris Farmer Union, start at .99 . I brought my at Fedco Seeds http://www.fedcoseeds.com/. They have great selection of seeds and the price are very reasonable.
Also checkout the following website: http://www.newenglandgardener.com/just-add-water/getting-an-early-start-on-the-garden-in-maine/
Happy Gardening!

Hello Anna;
Where are you? North or South? East or West? the climate is very different in these four parts of the state. I live in Saco and we have just started a community garden. Unfortunately, we have sold out all of our plots but you can attend the meetings and get advice if you are near by.