Lets say we have a satellite wanting to go to Mars. Do satellites such as MAVEN inject into a Hohmann transfer orbit while in Earth's orbit, or do they go out of Earth's Sphere of Influence into solar orbit, and then inject into a Hohmann transfer orbit? The reason I'm asking this is because it requires a long burn time to obtain the necessary $\Delta V$ and by that time, the satellite can be in the wrong position as satellites move fast in Earth's orbit (LEO specifically). However this can all be avoided while in solar orbit.
In general all spacecraft will try and do most of their acceleration as low in a gravity well as possible. This gets you the maximum benefit from your fuel, something referred to as the Oberth effect. For this reason an interplanetary mission will generally inject directly into transfer orbit to its (first) destination planet from Low Earth Orbit, or even directly from launch without entering LEO.
The trajectory design will include allowing for the actual time the rockets burn for and the movement of the spacecraft during that time. There is also generally an allowance for a few small trajectory corrections midflight.
A "Hohmann orbit" is an ellipse connecting orbits around the starting and destination objects, and in this case, it's an eccentric orbit around the Sun which touches (is tangent to) the initial Earth orbit and possibly the final Mars orbit.
You have to exceed Earth's escape velocity to even be in such an orbit, so the delta-v is the difference between your orbital velocity around the Earth and the transfer orbital velocity. Don't forget the different reference frames! (see answer(s) to How to best patch my conics?)
The technique of building up a plan from transferring from one conic orbit to another is called the patched conic approximation and has been a staple of trajectory design due to its simplicity, speed of computation and search for optimal design, and it usually provides relatively good initial accuracy.