Sometimes home storage sheds need to be accessed regularly and, despite being close to the house, are still a hassle to go outside and walk over to. When this is the case, it is often a good idea to attach the storage shed to the house itself as a sort of ancillary room, which you won't have to deal with going outside to reach.
By simply building a covered walkway from a back or side door of your house to the door of the shed, both structures will be effectively connected to each other, making the commute between the two easier to do in inclement weather. The pathway between the two structures should be covered with a hard surface like cement or gravel and mortar so that items with wheels can be easily wheeled up and down the path between the two areas. A metal roof with gutters should, ideally, also be installed, so that runoff doesn't destroy the plant life or landscaping which is growing around the newly covered pathway.
If you want to go the extra mile to connect your storage shed to your home, an actual hallway can be built between the storage shed and the house itself. By using the same basic structure as the covered walkway, a variety of materials can be used as walls in the hallway to protect from inclement weather and extreme cold between the house and the storage shed. This all depends on the length of the hallway and how much work you want to put into it. If you simply want to fully cover the passageway between the house and the storage shed, simple siding made from aluminium can be used. If you want some insulation in the hallway, a frame will have to be built and insulation installed with sheetrock and siding, so it will hold in heat and cold air. However, the more of a structure you build, the more expensive the project will become.
Depending on the reasons for why you want to connect the shed and the house, an additional room can be added on to the house itself. This will become a relatively large construction project and building permits will probably have to be acquired from the local building permit office. These additional rooms can be any number of things, but it may make the most sense for it to be a garage or shop, so that the additional room's style and decor merges well with a shed simply meant for storage. These additional rooms should have lighting and power and be attached seamlessly to the original structure of the house itself.