How fast should I lose weight?
The explanation of the question can be quite lengthy, so instead I’ll just give the general safe practice. The fastest speed that one can lose weight in a safe manner is roughly 1% of their total body weight per week. A more sustainable approach would probably be about 0.5% per week. Generally, this looks like 1-2 lbs of body weight per week. The faster you lose weight, the more muscle mass you will also lose. This means that while you lose 2 lbs per week, 0.5-1 lb of it may actually be muscle if done incorrectly.
To minimize muscle loss during weight loss, it is important to both 1) eat enough protein and 2) engage in resistance training. Eating protein will only slightly dull the loss on its own, while adding in resistance training will greatly reduce, and to some degree stop, muscle loss during weight loss. The slower the speed, the less muscle that is lost. In the end, losing weight and maintaining muscle mass is both a matter of resistance training and nutrition.
How fast should I gain weight?
This question is more goal dependent than the last one. Generally people don’t want to lose muscle intentionally, so losing weight makes sense. Gaining weight could have two different purposes though. For this reason, I’ll split it up into two different explanations. First, for those whose goal is only to gain weight. Second, for those who are gaining weight for the sake of putting on muscle.
I’m sure many reasons exist to put on weight for the sake of just getting bigger. Perhaps nursing back to health from various illnesses, or recovering from disorders. Either way, I am not aware of any major studies done on this particular subject but there are still general principles that can be applied. Changing the body drastically one way or another typically has negative consequences, so I assume as much to be true about weight gain as well. For a safe, and also probably more manageable, increase in weight, I would keep it to 1 lb per week. This is nothing more than my opinion, but I can imagine gaining weight too quickly would have negative effects on the skin.
Putting on weight for muscle is, however, something with plenty of research to guide it. The general consensus is that excess calories is not itself what causes muscle to grow in an individual during recovery from resistance training. A caloric surplus does, however, help maintain an environment where the body can utilize protein and excess energy to form new muscle mass as it repairs itself. Essentially, the body inhibits certain hormones from occurring when in a caloric deficit which makes it more difficult to build muscle.