Personally I would book my flights first. You will always be able to get decent accommodation at a reasonable price, but I wouldn't risk it with flights.
They are flight comparison websites that you can enter your preferred dates and they will bring up a list of flights available. Alternatively you can search directly with the airlines website.
You have not mentioned where you are flying from, but here are a list of possible airlines you could use to Dalaman.
Thomas Cook Airlines
Thomson Airways
Monarch Airlines
Viking Airlines (Kiss Flight)
Pegasus (Kiss Flight)
Onur Air (Goldtrail, H4U, Aegean Flights)
Turkuaz Airline
Easyjet
Bmi
Jet2.com
KTHY KIBRIS (Turkish-Cypus Airlines)
Turkish Airlines (Via Istanbul)
With the apartment, again you have a few options. You could book directly with them, if they have a website and accept bookings this way. Alternatively you could google search them and it will normally bring up a list of tour operators that you can book them with.
With regards to transfers, yes you can book them with a tour operator (if you choose to book the apartments with them) however this is not the cheapest option. If you are looking to do it on the cheap try some of these companies.
Prices start from as little as £14 pp return
transfer4cheap
raintransfer
transferonly
hope this helps!