Hi! There are 2 trekking seasons in Nepal for high elevation trekking, i.e. during Nepal spring (April-May) and autumn (September-October). Trekkers prefer autumn season just after monsoon, because the air is very clear and the clarity of panoramic Himalayan views is superb.

Climbers on the other hand prefer spring season because the snow conditions are better for climbing in spring season. Trekking on lower altitudes can be done any time outside the monsoon season and if you want to avoid crowds, trekking lower elevations between November and March is a good choice.

Seems you are interested in trekking rather than mountaineering. Though I am not an expert, i can suggest you as per my friend's experiences; to be very careful about your health issues at first. You have to make sure you have no altitude sickness and you are capable enough to make smart decisions at difficult circumstances and the most important thing is you must must must have a professional or experienced person to guide you throughout your trek so that you won't make another headline in a newspaper next morning. Hehehe! Good luck!