It isn't possible to book entrance into the Duomo or surrounding monuments (cathedral, baptistery, museum, tower, dome).
Check out the opening times here:
http://www.operaduomo.firenze.it/english/opera/orari.asp
Don't go exactly at opening hour since you'll already find a line forming outside - but the best time to visit would still be mid-morning or late afternoon.
I'd do either the dome or the tower but not both - you have over 400 stairs to climb for each and no elevator. Both give you wonderful views of Florence.
There are many companies out there offering tours of Florence in general but for the duomo and the other monuments in the Piazza del Duomo, I'd go straight to them (cathedral administration).
The OPA Centro Arte e Cultura offers daily tours as well as special visits that you normally can't access (the terrace of the duomo, for example, is open to the public only once a year (Sept 8) and gives you a different view of both the tower and the dome as well as a view over all of Florence).
There is some information here
http://www.operaduomo.firenze.it/cac/serv02.htm with times and costs for the daily tours - the information for now is only in Italian, unfortunately.
If you click on each monument, you'll see the cost - for example, the entrance fee plus guided tour for the Baptistery is 7 euro which is really good, considering the entrance ticket by itself is 4 euro.
Theoretically you can book online but I wasn't able to install the special plugin they require - probably easier to just email or call to get more details. It seems they also offer walking tours of the city center so it might be good to ask for more complete information.
artecultura@operaduomo.firenze.it
+39 055 282226
I've never encountered a line at the Bargello and just a short line at the Medici Chapels - you could book ahead paying the extra booking charge if you want the peace of mind of knowing you will skip any line. The official site for booking is:
http://www.b-ticket.com/b-ticket/uffizi/default.aspx
June is generally warm, so try to be inside during the hottest parts of the day. Definitely do the tower or dome early or late in the day as opening times permit.