When to go: The best seasons for trekking are either side of the monsoon season, March-June and September-November. During this time the weather is generally fine and the skies clear. It is possible to trek out of season, but expect lots of rain (and leeches) during the monsoon and severe cold and closed passes during the winter months.
Experience & Fitness: There are treks suitable for a wide range of experience and physical fitness, for age 5 to 85. An easy teahouse trek with Nepali support (guide/porter) is quite attainable for anyone who is reasonably walking fit - if you can walk for a few hours each day for a week and are not averse to the occasional (frequent!) hill climb then you can trek in Nepal. Longer treks, crossing high passes and into remote regions do tend to demand a higher degree of endurance. For Trekking Peaks it is usually desirable but not necessarily essential to have some alpine climbing experience.
Equipment: the main essentials are sturdy and comfortable hiking boot, a sleeping bag and a few clothes (be prepared for a range of weather). It is best to travel light, take only what you need and leave the rest behind. If you have the services of a porter then you will need a day-pack for your essentials and the rest goes in a kit-bag or duffel to be conveyed to your next stop. It is possible to buy everything in Kathmandu and Pokhara but it is all copies.
Hiring support: Whether to join a group, trek with other independent travelers or to hire your own guide and/or porter is a personal decision to be based on the difficulty of the trek and available budget. When signing up with an agency you should speak with several and make detailed enquiries about the differences in service besides just the base cost. If hiring staff independently then be mindful of your responsibilities to ensure that your man is suitably equipped for the job and stays safe.
Permits and TIMS: "Trekking permits" are not required for the main teahouse treks. Recent rule brought in by the Trekking Associations in Nepal require that all trekkers register with TIMS ("Trekking Information Management System"), this can be done via official trekking agencies or the Nepal Tourism Board. Trekking to remote areas and climbing the designated "Trekking peaks" require extra permits; these are generally obtained by the agent/guide who will be arranging your trek.
Arriving mid-travel: If you arrive in Kathmandu part way tour of Asia and decide to go trekking then you can easily get equipped in Kathmandu. Plenty of shops in Thamel sell (or rent) any trekking gear that may be required.
Rescue insurance: Before the departure check that your insurance covers trekking activities and the conditions. It would be very costly to pay a helicopter rescue at 5000 meters.