How quickly can I change my body and grow muscle?
Assuming a solid training program (emphasizing major compound exercises - squats, presses, deadlifts, rows, etc.), here's how it typically goes.
For the first 3–4 months, you will get stronger quickly, improving your capability on nearly every lift nearly every week - and that will happen mostly because you are simply learning to perform the exercises more efficiently, becoming less intimidated, learning how to push and pull hard, etc. - that fancy term for it is "neural adaptations." It just means learning the skill and developing the ability to better use the strength you already have. Your body will "change" - you will become stronger - and you may feel a little tighter and more solid, but you are not going to experience significant muscle gains in that initial novice phase.
After the first few weeks and months, if you keep progressively overloading your training - gradually lifting heavier weight and/or more total work - the wonderful early "neural adaptations" begin to peter out and your body will be forced to adapt to the continued demands you are putting on it with more adaptations to the tissue itself - in other words, the fibers that make up the part of your muscles that slide against each other and contract start to accumulate and grow protein structures that make them thicker (you might think of threads gradually turning into strings), your tendons attaching them to the bones grow denser and tougher, and you start to see physical changes in the size of the muscles you are training. Your clothes will start to fit differently and so forth. Depending on the quality and propensity of your unique body to build muscle tissue, the adequacy of your training stress, the quality and quantity of your diet and sleep, stress, etc., you will start to look and feel substantially different somewhere between 12–24 months into your training - your shirt and pants sizes will probably change, other people will notice, etc.
It takes 5–7 years of dedicated training (and, as always, good diet and recovery) for most people to begin "getting close" to their genetic potential. By that time, everyone will instantly know you lift weights at a glance. You will be the strongest and most "jacked" person in most any room or environment you are in, other than a gym full of other people who train too. It will become harder and harder to make gains in strength, size, or body composition (just because you are getting closer and closer to the best you can be), and at this point you will be working on weaknesses, details, etc. - the finish carpenter, no longer the framing carpenter.