I'm not entirely certain what you're asking - the use of the term 'soil' is confusing. What you buy at the garden centre will be either potting compost (John Innes, Multi purpose, but still potting) or soil conditioning compost, or composted animal manure, or topsoil. Topsoil is only as good as where its been extracted from - its dug out of the ground and bagged up, so sometimes it might be rubbish, like motorway spoil, or it might be top graded soil.
The 'soil' you speak of with added nutrients and peat, etc., is a good description of a general potting compost rather than soil itself.
So, if you're asking if you can improve the soil you've already got, yes is the answer - humus rich materials are what you need, and by that I mean good garden compost you've made yourself, soil conditioning compost or composted animal manures that you buy in, plus things like leaf mould, spent mushroom compost, etc.
If you're asking if you can make your own potting compost, you can, but your own garden compost would need to have been produced using a 'hot' aerobic system rather than the more usual cold, anaerobic one because the heat is what kills off weed seeds and certain other undesirables. To that you'd add a quantity of sand and peat, maybe some grit; there are many recipes for making different types of potting compost, depending which plants you want to grow in it.