The easiest and most economical way of getting to Asia from anywhere in the West is to fly, though you can reach the region by other means too, like by train or boat. Obviously this will take quite a bit longer.
The vast majority of people reach South East Asia via a flight to either Bangkok or Singapore, both of which can be flown to direct from London. The capital of Malaysia, Kuala Lumpur is also a popular entry point, as is Jakarta, the capital of Indonesia. Other major cities which receive international flights are Hanoi, capital of Vietnam, Phnom Penh, capital of Cambodia, and Phuket, the largest of the Thai islands.
For elsewhere in Asia, it's usually easy to fly into any major city. Most flights international flights into China arrive at Beijing or Shanghai. and it's very easy to fly into Hong Kong. South Korea is served by an airport just outside capital city Seoul, and most flights to Kazakhstan arrive at Almaty. Have a good look around when booking flights, and you shouldn't have any trouble.
From the West, journey times can be lengthy. For example, a flight from the UK to Bangkok is around 12 hours direct. If you opt for a cheaper fare to Asia it will usually include a stopover, usually somewhere like Dubai, for a few hours. This can add a significant chunk to your journey time, but is kind to your budget.
It’s possible to reach Asia by train. You can actually travel all the way to China from Europe by train, via the Trans-Siberian Express from Moscow, which is an epic travelling experience in its own right and certainly makes for a better story than hopping on a plane. However it is more expensive than flying in the long run and will obviously take considerably longer. From there it's easy to get a train into Vietnam and beyond.
It’s possible to reach South East Asia via the sea, but this is reserved almost exclusively for those on cruise ships, so if this doesn’t feature in your gap year plans then you will need to consider one of the above two options.