If you haven't got a spiritual teacher (and I suspect this is most of us!) then you could do a whole lot worse than befriend a tree.
In reality there isn't much information written and/or readily available on tree spirits (often mistakenly called dryads - the original meaning of 'dryad' referred to the spirit of an oak tree). This means that most of your work with tree spirits will have to be guided by your intuition. If you were looking for hard and fast instructions, sorry folks!
Befriending a tree will bring a whole new quality to your spirituality and will certainly add your connection to the spirit of the land. Trees are so very different from humanity and there is so, so much that we can learn from them.
How do you find your friend then? Are there any particular trees that seem to reach out to you - either spiritually or literally as you walk past they always seem to brush you with their branches? Is there a particular species that you feel an affinity with? These are good questions when searching for a tree to befriend. If you're lucky you have a tree growing right in your garden and it is already part of the family. If not, there are trees even in the most urban of situations.
I would start off by just showing the tree kindness. Water it during drought. Collect any rubbish that has been left at its base. Sit with the tree and give it time to get to know you - trees don't rush! Slowly you should start to develop a connection with the tree, and then who knows what it will teach you...