It depends on the airline and its particular situation and sometimes believe it or not in the very same laws of market...
Some airlines, as stated above will give you an option; nevertheless that hardly happens, at least on this side of the Ocean. Upon a variety of aircraft, those with more hours will get a bigger or more complex aircraft. The ones with the minimum commercial hours -if they are lucky enough to get hired or have the right strings at least- will spend a few years in the right seat of a turboprop or in the smallest, sometimes oldest type (i.e. a brand new EMB145 will be flown by those with more experience or seniority than those flying lets say... a 737-200).
In my country.... The main airlines work with a very powerful Pilot's Union. Once the recently graduated pilots affiliates to the Union and enlist themselves in the airline of their choice, then wait SEVERAL months if not years to receive a "Letter" from the Union to participate in an admission test. If you do well and get admitted to the airline, you will go to the right seat of the smallest of their aircraft.
In example... If joined AeroMexico; as of now, the career path should be like this:
First Officer Saab340 (AeroLitoral)
First Officer EMB145
First Officer MD80 (AeroMexico)
First Officer 757/767
First Officer 737-700
Now... Remember what I said about the "market laws"? Here is why...
In many places in America, including the US, Mexico, Argentina and many other countries, the smaller non-unionized airlines will hire pilots if they hold the type certificate of the airliners they use! The airlines no longer have to worry about giving expensive "initial" training. As there are lots and lots of pilots without a job, the airlines are in position to choose and be very selective among a large quantity of candidates. The better prepared, the ones with the appropriate rating will surely get the job and that is why more and more aviators are paying their own independent type rating training (happens a lot in Executive/general aviation and with Boeing 727/737-200-300) in order to be eligible.
There are no stupid questions... just stupid people!