To control slugs, grab a torch and try catching the slugs at dusk or later, especially in wet conditions which bring them out more. Don't touch these sticky creatures, rather scoop them up with something and toss them in the compost, in a far corner, out on the road verge, or if you're fed up, drown them in salty water or stamp on them — did I say that? Also make a trap with beer or sugar water, see Organic garden pest control.
You can't get rid of mushrooms in a garden, nor should you want to. Mushrooms are fungi and some help plants roots absorb moisture and nutrients, and some help with decomposition of plant matter. The mushrooms you see on the soil surface are only the fruiting bodies which hold spores, but underneath the soil is where the mushrooms' tiny threadlike cells called mycelium are, doing all the good work.
Mushrooms in a garden usually grow where trees are. There are 1000's of species, mostly harmless, and the ones that are poisonous have to be eaten to affect you, so just touching them is OK, but of course wash your hands after gardening anyway. ~ Megan