How to gain muscles and be fit with low testosterone?

You do not have "low testosterone" and even if you did, you would build muscle and overall fitness the same way that everyone else does - through training. 350 ng/dL is within a normal range for a 35 year old man, for reference 450 ng/dL is the 50th percentile (the exact middle, neither high nor low).

Getting to a healthy bodyfat level will likely improve testosterone levels, being too high or too low will decrease testosterone however. Based purely off of observation, I would imagine that 10-20% bodyfat for a male is the range that you'd want to be in. To that end, your nutrition and training can help to improve testosterone if you aren't within that bodyfat range. However, changing your testosterone levels within a normal range isn't going to change much.

However, I would strongly urge you to not worry about it. If you really want to do TRT, that's fine if your doctor thinks it's fine. It would likely improve your capacity to build muscle and get fit, but you still have to put in the work to make it happen. TRT isn't going to provide the stimulus that causes your body to adapt towards building muscle and being fit. Instead, TRT will simply make such efforts more effective.

What should you do then? Focus the majority of your training efforts on big compound lifts like deadlifts, squats, and bench press. Including things like overhead press, lat pulldown/pullups, and rows would help to ensure everything is being worked. Include some cardio training if you want to improve cardiovascular fitness and burn extra calories. For nutrition, eat an appropriate amount of calories (start with your TDEE and adjust from there) and get enough protein (2.2g per kg of lean mass) and you'll be on the right track.

For further reading, I would recommend the following article. https://examine.com/nutrition/increase-testosterone-naturally/

Low testosterone in men is only caused by two main factors, there are actually more possible things happening here but the two most common are being fat and old.

Some people look skinny, even though they are fat. You can be a stick man of 60 kilos ( 144 pounds) but 25% bodyfat...which means you are borderline obese. Obesity is not about how much you weight but about how much of you is pure fat.

The other factor is age. In the first case you can just lose weight and have normal testosterone. In the second factor a medical doctor can prescribe you testosterone. It is legal and safe and healthy if monitored by a doctor, after all you are not taking excess testosterone which can cause various problems, but you are just restoring your normal healthy amount.

But what if you want to avoid injectable testosterone and being reliant on drugs? Guess what?! Women have 15 times less testosterone than men and they can build muscle up 60% or 70% as fast as males.

And unless you are a woman or your testicles were removed surgically, then there's no way you have such low testosterone.

So women can build muscle with 15 (ng/dL) of testosterone, not as fast as men but half as fast or slightly faster than half the speed of a man. You have 350 (ng/dL), that's way more than enough.

After the age of 20 males lose testosterone every single passing year, but they are never gonna get to the level of a female.

I have low T also (390) and have been able to gain muscle by training hard 3x a week (5 sets, 6 reps of bench, press, squats, pull downs, dead lifts). The key, however, it eating a ton (4 protein shakes a day with 30-50 grams of protein, along with three big meals a day). Do this and I GUARANTEE you that you’ll gain muscle, but you HAVE to train three days a week CONSISTENTLY! DONT MISS A WORKOUT!!!