You can certainly build a shed the way you are describing. Simpson (and competitors) make an entire line of brackets and connectors to connect essentially any wooden structural element to another wooden element, or to a concrete post as you describe. Go to the Simpson website, you can download free catalogs and software to help you select the proper connector.
For the type of structure you describe, typically you install beams from post to post, attaching the beams to the concrete posts using U brackets that are J bolted to the concrete. The joists typically sit on top of the beams, attached using joist clips. Or you can attach the joists to the beams using joist hangers. It all depends on your design. If you do not have a printed design, you may want to purchase a shed design book, or get some plans, typically the plans will include full details on how to build the foundation, and will show alternative foundation types such as posts or slab.