Well the obvious one is a lawn mower. Depending on the size of the yard, you will need either a push mower or a riding one. A weed eater can help you get the edges of the yard and any other small areas. A hedge trimmer will help keep your bushes looking nice.
The only machines that you should need to purchase for your yard are a lawnmower and weed eater. These will be tools that you will use most often. A lawnmower you will need to cut the grass in your yard and weed eater to get the grass around the house. You can rent larger tools for larger jobs.
First you will need a lawn mower, snow blower and edger. As a homeowner these are a must. Then you will need pruning shears, rake, shovel and hoe is you are planting flowers or vegetables. And buy a pair of good fitting gloves and a small hand shovel.
Here in Minnesota the average homeowner needs to have a lawnmower, fertilizer spreader, snowblower, chainsaw, tiller, edger, leave blower or a leaf bagger, and a sprayer for herbicides or insecticides.