It is a beautiful garden the previous owners really liked ornamentals. That being said, it is now your garden and you can do what you want. Several pictures are of Herb Robert which is a weedy geranium but has pretty flowers. It might be a little difficult to eliminate but it pulls up fairly easily. The plant that will give you the most difficulty is Indian Mock Strawberry (Duchesnea indica). It's the one that has leaflets in 3s and creeps along the ground. It's those creeping stems that are the hardest to remove. You can spray with Roundup or pull by hand for a while, your choice. The rest of the plants should be easy to remove with a shovel and probably won't come back
