As a new character, you start with your racial Frigate skill at 2. This gives you the ability to fly all but one of the Tech 1 frigates for your race, a total of 6 ships. Training the frigate skill to 3 (which you should do at the earliest opportunity) will give you access to the seventh ship. Each of the four races has a different fleet of ships with their own strengths and weaknesses. Today, I'll focus on the Gallente frigates.
Velator: The rookie ship. Gets bonus to damage. No matter what role
you can think of, there is another Gallente frigate that fills it
better. You have a free and inexhaustible supply of Velators, however. The same cannot be said of the
Navitas: This is a hybrid combat/industrial ship. Can fit decently for
combat, but gets bonuses to cargo capacity and mining yield. Also, has
a remarkably long locking range but, with frigate sized weapons, it is
difficult to take advantage of this.
Atron: Nimble combat ship. Gets bonuses to damage and range. Can't put
out or take as much damage as the Tristan or Incursus, but is the
fastest Tech I Gallente ship of any class.
Maulus: Electronic warfare ship. Gets bonuses to damage and to sensor
dampener effectiveness (dampeners reduce enemy range and increase
enemy lock time). Has a lot of CPU output and decent powergrid, and so
can fit the most varied and esoteric modules.
Imicus: Scout ship. Gets bonuses to scan strength and drone control
range. An ideal scout, but can make a surprising effective combat ship
with good enough drone skills. Is, however, frigging ugly.
Incursus: Combat ship. Gets bonuses to damage and range. This is the
primary Gallente brawler frigate. Can put out quite a lot of damage
and is relatively fast.
Tristan: Combat ship. Gets bonuses to damage and range. Has both the
most hit points and the highest raw theoretical damage output of any
Gallente frigate. However it's also the hardest to use because it's
damage potential is split between three different weapon types
(turrets, drones and missiles) all of which have their own skill tree.
Also has the problem of being unusually slow for a frigate, which can
be a thorny issue when you consider that to maximize its damage output
the Tristan needs to fit short range guns.
Every one of these ships has a practical or tactical situation for which it is the correct choice. Yes, even the Velator:
However, on your first day, soloing the tutorial missions, your goal is going to be to get out of the Velator and into an Atron as soon as possible. They only cost about twenty thousand Isk and you should have banked that much from running the very first tutorial mission. Take the plunge. Nothing beats that new Atron smell. The Atron is more than capable of handling everything the tutorial missions will throw at you, but you should move on to either the Incursus or Tristan as soon as you can afford to. You'll find both the Incursus and Tristan to be very capable solo combat ships, and their lethality will only increase as your skills (both as a character and a player) develop.
Even if you aren't playing a Gallente character, the basic frigate progression is the same. A Caldari rookie running the tutorial missions should hop into a Condor immediately and then move on to either a Kestrel or Merlin. An Amarr rookie will go Impairor -> Executioner -> Inquisitor/Punisher. A Minmatar rookie, Reaper -> Slasher -> Breacher/Rifter.
One thing to keep in mind
though that is that, while the ships of the different races can be quite
different from one another, there are no intrinsic differences between
characters of different races except that they start with a few days
worth of skills in the ships and preferred weapons of their own race.
There is no
"racial bonus" or anything for a Gallente character flying gallente
ships. In fact, if you created a Gallente character and spent three
days training Amarr starting skills and simultaneously created an
Amarr character and spent three days training Gallente starting skills
you would, in the end, have two characters that were functionally
identical in everything except for mugshot and what ethnicity they
tick on the census form.