It sounds like you want to backpack, and not just day hike. That's fine, except that you will need to get a wilderness permit. For days like Thursday through Sunday, there is the maximum demand for permits. You might be able to get a walk-up permit, but a permit reservation is slightly more foolproof, even if it costs a few bucks. Your time is valuable. Online at the Yosemite site there is a map of the park that shows which trails are managed by which permit stations, but the permit reservation office covers everything.
It also sounds like you don't intend to have any real lodging in the park. That's fine, and you won't be limited to any specific areas of the park.
What are your limits or preferences? Do you want to do five miles per day, or twenty miles per day?
This has been a fairly dry winter, so there isn't so much snow up high. OTOH, this is still a bit early, and the trail clearing parties haven't covered too much. So, once you get a few ideas, you can contact the permit reservation office. Often they can see the trail reports and warn you of a bad situation.
Also, you probably won't get anybody to commit to long-range weather predictions. Even though the scant snow has melted, it may not be to full summer weather yet. You need to prepare for an afternoon thundershower every day.
You will need to have a bear canister. Most of the time, the permit stations can rent you one, but that means that you have to return it to the same station afterwards. Yosemite black bears are thought to be pretty crafty.