Kitchens – Are there simple ways to construct shelving under a kitchen worktable

I like the enthusiasm, that's great. A couple suggestions: 1. Start off with a solid design, including what sort of materials might work. Let the wood material options drive some of the design which would help to save costs.
Explore the wood section at the depot and see what is offered.
Wood can get expensive. So let the options of wood drive the design.

1. Break down the pieces into their own distinct unit and determine the best way to connect them. It sounds to me you want a lot in your first project, so break it into pieces and build the breakfast bar first. This way the project wont take weeks to complete. You build the bar, it gets installed and you can work on the other pieces.

2. Decide the finish, if you plan to paint, the quality of the wood can be less as it will not be seen, however if you plan to stain, the quality needs to be higher. For example a 2x4 can be painted and look great, but try to stain over the ink stamps and its awful.

As for legs, be creative, since you don't have any saws, get a hand saw at least, and be creative. I bought a bunch of 1 inch square 36 inch in length pieces for some furniture pieces I was building. They worked great as the legs and were cheap, and already precut.

Final suggestion is to go slow, as this is your first project take your time, plan accordingly and have fun!

If the wood is dented rather than cut/gouged, you can steam dents out with a clothes iron (preferably a steam iron, but not required if you keep the washcloth damp), a damp washcloth, and some patience.

If it's actually cherry, and not just "whatever, with stain, called cherry since we used a reddish brown stain" it is best to remove and not replace the stain. It will start rather light, but age beautifully with time. But that is a matter of opinion. Your best bet with re-staining it is to take the whole thing back to wood, and re-stain the whole thing if you are going to do that - then it will all match. However, some slight mismatch between the top and the rest of the piece will not be as glaring as a mismatch between two halves of the same part. Try to find some similar scrap and test your finish process and colors on that.