I could write a book in response to your questions. We are embarking on year 6 in our house, and the last of the big yard projects (patio cover and new exterior French door) will be completed this year. How long it takes is a function of how big a space you have and how ambitious your plans are. We have about a quarter acre and had very complicated plans - including a vegetable garden, blueberries, a grape arbor, chicken coop, compost bins, gas fire pit, raised beds, gravel paths, 3 water features, 14 trees, hundreds (literally) of ornamental plants, trellising, hops arbor, etc.
For your project, I would recommend thinking about what your goals are and also your appetite for ongoing maintenance. Do you need outdoor entertaining space? Do you want a neat and simple front yard that matches the neighborhood - or something more complex or eye-catching? Do you need to hide any eyesores or add screening for privacy? Once you really dig into what you want to achieve from the space, you can develop a plan. The plan can then be divided into phases - depending on your time and budget. For example, if you need a new front walk to your front door, you may want to prioritize that first over buying new plants since plants and lawn would likely be damaged when installing new hardscaping.
A couple other thoughts:
When cleaning up, don't get too crazy - especially since you've only lived in the house since the fall. You might find that the tree you removed would have been great for shading the living room window in the summer or had really pretty spring flowers. I would advise trying to keep any healthy, well-sited mature trees currently on the property. They do wonders for making a house feel part of the landscape, and young trees seem to take forever to get big. We didn't have any mature trees on our property other than a rotten apple tree that had to be removed. It has been a serious bummer waiting for the new trees to get big enough to cast a little shade in the summer.
In your shoes, I would start with the clean-up and fence repairs. Digging out old shrubs is hard manual work but not something I would hire out. Just make sure the ground is moist and you have a Pulaski for hacking out the root ball. You should be able to learn everything you need to know about fence building and repairs with some googling.
For plants, I always recommend buying from a local nursery. The plants are going to be in better shape and the staff far more knowledgeable about what will work in your area. They're not necessarily going to be the cheapest, but I think it's worth the premium. For cheap plants, a lot of garden clubs and other garden-related groups (e.g. Master Gardeners, etc.) have plant sales in the spring. They can be a great way to get cool plants for great prices. For mulch, compost, and gravel, we buy from a local landscape supply yard. Hands down cheaper than buying individual bags at the home center.
Well, that's just the tip of the iceberg. Keep asking questions and reading. You might just find yourself with a new hobby.
