There is easy access to Lashio by bus from Mandalay or Yangon on Shwe Mandalar bus lines. 15,000 Kyat ($15 from Yangon). This bus is comfortable. You can upgrade to 22,000 Kyat ($22 from Yangon) for VIP class bus. You can buy bus tickets online by searching for Myanmar Bus Tickets, and the site I used I was able to pay with Paypal.
There are also vans that travel once they are full. No info on that option, as I've only heard of it, not used it.
You can get there from Hsipaw (locals pronounce: Thibaw) with the train which departs mid afternoon, and costs 600 Kyats per ticket. Bear in mind that the train supposed to arrive in Lashio at 7.30 or so but you will arrive at least at 10pm in a border town where few people speak English. You can do a full day train, but it is a long day from Mandalay all the way to Lashio. But there are good places to stop for a night or several nights along the train journey. Pyin Oo Lwin (Maymyo) is known for its cool weather, beauty, and colonial era influences. The train goes from Pyin Oo Lwin over the Gokteik Viaduct bridge and a fitting stay for that night would be Kyaukme. There are a few trekking guides there who can take on day trips of hiking or motorbiking, or multi-day treks. Similarly, a few stops down the track is Hsipaw, which is similarly beautiful but has an overcrowded tourism industry. The next main stop past Hsipaw is Lashio, but the most beautiful part of the train journey, as you wind along the river, passing a couple beautiful waterfalls.
Air. You can fly to/from Mandalay, Tachileik, Yangon, Heho (Inle / Kalaw), or nearly anywhere in country. Lashio airport is an old military airstrip, as are almost all airports in Myanmar aside from Yangon and Mandalay. It is a typical small Myanmar airport, as it is just a concrete small building with a few wood benches in a middle of nowhere. No computer or internet or screen of departures or English signs or even a phone. There will be a few small shops selling some drinks and snacks. Someone will shout the name of the company you are flying with. Go there with your tickets printed or an email of your booking, and arrive at the airport earlier than you think (our fight departed 25 min early)
At the bus stations or stops and at some travel agents or hotels, you can arrange a private taxi to Lashio. This allows flexibility to stop if you negotiate this into the price. This is more expensive than bus or train, but is also more comfortable and convenient. If you want to ensure that the taxi is all yours, you need to pay for all the seats, otherwise they will want to fill it with other people.