Congratulations on your new home! Mature fruit trees can be pruned to make them more productive and easier to harvest, but it is usually done over time in stages so as not to shock the tree. Here's some sites with general info and info on how to do this.
Be sure to scroll down to 'Pruning Neglected Apple Trees' as well as reading the entire page here.
This site has some great pics.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.