The slayer skills (Mindread, weapon expertise) stay at level 1 when you unlearn your skills, but I don't recall what happens with the class skill bonus. Maybe I can check that later.
Skill upgrades are instant, though they can have level requirements. The cost depends on the skill and which trainer you have, with the first upgrade to level 10 costing either 750 or 1500 gold, and the second upgrade double that (1500 or 3000, respectively).
You can use the skill trainer right away. There are platform upgrades that require quests, which can make the platforms more effective. You may need to upgrade the trainer's platform to get the second stage of upgrades.
