Welcome to the forum plasbart 

With using the vegetable skins ,I think it depends on a few things , how old the vegetables are and what is being cooked.
New spring or summer veg  are usually tastiest with their young fresh skins , they are soft and not unpleasant to eat .

But later in the year and with more a
Mature vegetables often the skins are tough and unpleasant to eat .
Winter veg , things like swede have tough skins and the skins are are not great to eat.

With garlic , it depends on how you are using it . If its being baked , roasted , used in stocks then there is little point in peeling it . The skins will help protect the little clove . The papery outer casing is not usually eaten , once cooked you just ease out the soft delicate flavoured cooked cloves (and they are a joy to eat ).
If I am using galic in something like stock then I will just cut the blob in half or if I don't want so much flavour , I just bash it to split he papery skin and just crush the clove a bit ( usually with the base of the olive oil bottle as its handy !) I then add the clove papery skin and all .

If you are using a garlic crusher then often you can just out the unpeeled clove of garlic into the crusher , you need a little bit of extra muscle power but the clove will generally still crush and the skin gets left with the debris

As far as I know , here are generally considered to be more nutrients closer to the skin of the vegetable . If I am peeling vegetables then I only take off the depth of peel that I feel is needed to make the vegetable more palatable for the dish I plan to cook .

The peelings that I take off don't generally get wasted ... They either get added to some stock I am making ...or go into the composter to rot down and eventually be fed back onto the garden !