Launching from Florida 28.5 degree inclination to rendezvous with ISS at 400km to 51 degree inclination of ISS
The latitude of a launch site determines the minimum inclination that can be directly reached; launching from 28.5º latitude in the due-East direction achieves a 28.5º orbital inclination. Launching to any higher inclination is straightforward, simply by steering continuously North-of-East during the ascent. In the extreme case, if you fly due North, you enter a near-polar orbit (the rotation velocity you receive from Earth before you leave skews things a bit).
In the case of the space shuttle:
These two images show that the flame trenches at KSC point due south, and that the Orbiter's tail when installed on the pad also points due south.
(source: Google Maps)
Therefore if the shuttle launched and went into a gravity turn in the desired heads-down attitude without rolling at all, it would have flown due south.
As explained in this answer, a launcher gains the maximum benefit from the Earth's rotational speed if it launches due east. This will place the vehicle into an orbit with an inclination equal to the launch site's latitude - in the case of Kennedy Space Center, 28.5º.
To fly due east and obtain this benefit while maintaining the desired heads-down attitude, it is clear that the shuttle had to roll 90º since it was mounted on the launch pad with the tail pointing south.
However, with the inclusion of Russia in the International Space Station (ISS) project, an orbit inclined 51.6º was chosen to allow the use of Russian launch sites.
For the shuttle to reach this 51.6º inclination, it had to roll even further north to align with this flight azimuth. Therefore, the vehicle performs a single axis rotation (SAR) maneuver, often called the "roll program", after clearing the launch pad, to align its direction of flight with the desired azimuth. This SAR both aligns the flight path with the desired azimuth, turns the vehicle heads-down, and starts the gravity turn.
To minimize the amount of delta-v required to change orbital planes, the launch time for an Shuttle mission to the ISS was set, if possible, to the time at which the ISS orbit passed over the launch site. However, choosing this launch time did not preclude the necessity of rolling to the proper azimuth. If the shuttle had launched at the exact time of ISS orbital plane passage with the desired heads-down attitude and not rolled, it would have flown due south.
In this video taken from a camera on the right-hand Solid Rocket Booster of STS-127, you can clearly see the vehicle roll from tail-south to tail-east. In the video, liftoff occurs at ~0:50 seconds, the roll starts at ~1:00, and at ~1:17 you can clearly see the coastline under the tail.
The only real implication of the latitude is the initial velocity of the rocket, and as long as the latitude is less then the inclination of the orbit. So its the orbit inclination that matters and constrains the latitude, not the reverse.
First the rockets close to the equator will start with highest velocity for simple reasons that we are on spinning oblate spheroid, the equator regions have the greatest linear velocity even though the whole thing has the same angular velocity.
As to orbit inclinations, the iss has larger inclination then Florida latitude, thus they time the launch till roughly Florida intersects the plane of the orbit. As you can see orbits of lesser inclination of Florida (equatorial orbit) are tougher and it would be preferred if you launch from equator. Thus launch pads at equator are ideal and most versatile except for high inclination orbits; in which case the initial velocity hurts and adds to delta v. But for normal non polar, non retro grade orbits, equatorial launch pads will require less delta v.
They do not "roll north" or anything as much as they crucially time the launch to orbital plane intersection. Then they burn tangential and in orbit plane (to constrain optimal vector with out considering atmosphere). This is the optimal trajectory to orbit with out considering atmosphere. To raise altitude is best done in small burns or several burns at roughly periapsis. Depending on which of the two intersecting points and the time of day chosen, it could be they travel southwards, not north.
Usually they will leave the orbit altitude lower or higher then iss, and again time so the two vehicles orbit periods are different, and then they will come closer and closer for docking.
If they cannot time launch site/orbit plane intersection due to the latitude or inclination, the rocket must first get to a point on the orbit plane before it can start building the energy required to achieve that orbit. It can combine the maneuvers but then it still would need to null the velocity tangential to the equator. While velocity parallel to the equator is still used.
In general points I like to stress are orbit inclination changes are hard and costly, but this isn't needed if by launching from site in the orbit plane, and this is done by timing the launch to the ISS or for vehicles already in orbit; timing burn to the point where both vehicles orbits intersect. Thus at best there are only one launch window every 12 hours to the ISS, but there could be additional constraints narrowing the launch Windows.