First aid is a secondary profession (along with fishing and cooking). You may have all 3 secondary professions as well as 2 primaries. In other words, first aid is a freebie everyone can have.
As for your primaries- if you take blacksmithing, you need mining. Mining will get you the raw materials to make into weapons and armor. Without it, you'll need to buy them all from other players. Blacksmithing/alchemy would be particularly bad, as the raw materials for alchemy come from herbalism which you won't have either.
Generally, if you want to make stuff, you need to take professions in pairs. Mining/blacksmith, Mining/engineering, skinning/leatherwork, herbalism/alchemy. The only trade skills that do not have an associated gathering profesion are tailoring and enchantment.