Can I use 10 gauge to first distribution point, then 12?
You haven't said how you will manage the transition from 10 gauge to 12 gauge.  If you are thinking of wire nuts in a junction box, then the answer is that you cannot do this safely.  The 20A breaker's job is to protect the load wire connected to it.  A 12 gauge wire needs a 20A breaker to protect it.  You cannot guarantee that a device at the end of one of these runs will not draw in excess of 20A (the rated capacity for 12 gauge wire).  Your larger breaker (30A, the maximum 10 gauge is rated for) will not trip in this situation and the 12 gauge wire may start a fire.  In addition to being unsafe, it is also not to code.  Code requires the breaker to be no larger than the ampacity of the wire and since you're using 12 gauge wire at some point that's your limiting factor.

However, if you are thinking of installing a sub-panel then you may be able to do this.  It would be fine (if you follow all the rules for sub-panels) to run 10 gauge wire protected by a 30A breaker to a sub-panel and then run 12 gauge wires each protected by a 20A breaker from the sub-panel.  However, this seems a little silly (if you're going to all this work you probably want to run more than 10 gauge to the sub-panel).

If you explain more about your goals you might get more useful advice about what approach to take.