Sitting on the Terai (the lowlands bordering India) and about equidistant from both Kathmandu and Pokhara, Chitwan National Park is easy to reach from both cities. In fact, it's Nepal's most popular jungle park partly due to its accessibility.
The information here is assuming that you'll be wanting to get to Chitwan from Kathmandu, Pokhara, or Bandipur, which are the most common routes. While it is possible to get to Chitwan from other parts of Nepal, doing so will usually require going through Pokhara or Kathmandu, taking a private transfer, or riding uncomfortable local buses.
It's also important to be aware of the different villages in Chitwan that you can base yourself in. Most visitors stay in or around Sauraha, on the eastern edge of the park, which has many hotels and tour operators. Others stay in the quieter village of Barauli, in the west, where there are a handful of accommodation options, including the high-end Tiger Tops Tharu Lodge. It's important to know whether you're staying in Sauraha or Barauli when planning travel to Chitwan. Many people will assume you're staying in Sauraha.
From Kathmandu
By Air
Duration: 20 minutes
The city of Bharatpur is the air hub for the Chitwan National Park, about 10 miles (16 kilometers) from Sauraha, or 30 miles (48 kilometers) from Barauli. There are several daily flights between Kathmandu and Bharatpur, which take only 20 minutes once you're in the air. It's not an especially scenic flight as you're going away from the mountains. Flying is by far the easiest and quickest way of getting between Kathmandu and Chitwan. Taxis can be found easily at Bharatpur Airport.
By Private Transfer
Duration: 5+ hours
The Chitwan National Park is about 60 miles (96 kilometers) from Kathmandu, and on a good day this journey may only take about five hours. However, road works leaving Kathmandu and along the Prithvi Highway, as well as traffic jams, can slow down the trip by many hours. It is a scenic trip though, following the Trisuli Valley much of the way, with its steep forest-covered hills and farming villages. The road turns south at Mugling and heads towards the Terai.
Many hotels and travel companies can arrange private transfers between Kathmandu and Chitwan. This is the most comfortable way to travel overland. Private transfers aren't much faster than buses, but they do usually come with air conditioning, good suspension, and the ability to stop when you wish.
By Tourist Bus
Duration: 5+ hours
Tourist buses are another way to get from Kathmandu to Chitwan. These leave from near Thamel early in the morning and arrive in Sauraha five or more hours. They usually make a few comfort and meal stops, although not on demand. Tourist buses are usually air-conditioned, and moderately comfortable. Local buses are also available, but are generally uncomfortable, crowded, and not as safe as tourist buses.
From Pokhara
By Air
Duration: 10 minutes
There’s one flight daily between Pokhara and Bharatpur, and it's very short, so could potentially save you a lot of time going overland. In the peak seasons, book as far in advance as possible as this single flight fills up fast.
By Private Transfer
Duration: 5+ hours
Chitwan is 52 miles (84 kilometers) from Pokhara. Travel specialists and some hotels can arrange private transfers. The journey from Pokhara is less likely to be delayed by enormous traffic jams, as delays around Pokhara aren't usually of the same magnitude as those around Kathmandu.
By Tourist Bus
Duration: 5+ hours
Most tourist buses leave from the Tourist Bus Park in Pokhara, although the more expensive and higher quality Green Line Bus leaves from its own stop near Damside. They arrive at the Tourist Bus Park (Bachhauli) just north of Sauraha, and make reasonably frequent comfort and meal stops. If staying in Barauli, it’s probably faster to get a tourist bus to Sauraha first and then get a private transfer (or local bus) from there.
From Bandipur
By Private Transfer
Duration: 3+ hours
Many travelers book a private transfer to take them from Bandipur to Chitwan. These can be arranged through hotels or independently.
Bandipur is a couple of miles up a steep hill from Dumre, on the Prithvi Highway. Private transfers will be able to pick you up directly from Bandipur, without you having to make your way down to the highway first.
From Dumre, you'll first travel east in the direction of Kathmandu and then turn south towards the Terai at Mugling.
By Bus
Duration: 3+ hours
All buses traveling between Pokhara and Chitwan can stop in Dumre on request. Travelers can pre-book a tourist bus from an agent in Kathmandu or Pokhara, who will advise of the best time to stand on the highway and wait for your pick-up. If you have a local cell phone number, you can track the progress of the bus so you don't wait for too long at the side of the road for a bus that's running late. Hotels will normally be happy to make the call for you if you don't have your own phone.
From Bandipur, most accommodation can arrange for a private transfer down to Dumre so you can meet your bus, possibly for a small fee. Local buses also leave Bandipur periodically, from the end of the road at the entrance to the town.
You may also be able to flag down a cheap local bus traveling to Chitwan, but these are not very comfortable or safe, and don't always run on predictable schedules.