Before you plant anything at all, I would take some time to find out and think about:
How much time you realistically will be able to spend maintaining your garden.
Which style of garden you prefer, and the essential features you need - it's important to match this to your time available. Don't choose a high maintenance style with minimal time. You will end up feeling overwhelemed with no time to sit, relax and enjoy your efforts.
What type of soil you have.
The climate in your part of the country,
Which part of your garden is in sun/shade during the day.
Decide on your garden design and style before starting any work.
This Youtube video from the Middle Sized Garden is very useful in showcasing different styles of garden -
Although many of these gardens will be larger than yours, the styles will give you a starting point for design.
Use ropes and bamboo canes to mark out where your proposed features will be located, such as patios, arbours, planting beds.This will give you an idea of scale and balance in the garden.
Once you have sorted out the hard landscaping features, look to the type of soil, climate, and the sun/shade areas. This will determine the plants that will thrive there.
Make yourself a sun/shade sketch making note of how many hours sunlight each part of the garden receives. Does the area at the end of the garden get much sun or is it always shaded by the fence as the garden is south-facing?
Palm trees - some of which are listed as hardy in the UK -might only be so in the warmer southern areas. Some palms need certain types of soil, have specific sunlight requirements; some will grow to over 12 metres.
Take time to plan your garden - it will enhance your enjoyment and utility, and save costly mistakes.