How you train new spells and how you gain new power points are resolved in different ways.
I have at star Magic 4. i have 8 spells.
Right.
With magic or resonance A as a full magician you will have an unadjusted magic rating of 4 which mean you get 8 spells or rituals for free. This only apply to full magicians (not adepts, mystic adepts, aspected magicians or technomancers).
For example Improved Invisibility + Healing + Elemental Armor + Increase Attribute + Levitate + Lightning Bolt + Power Bolt.
As adept i have Power points 4 and i can purshase Adept powers to 4.
Right.
With magic or resonance A as an Adept you will have an unadjusted magic rating of 4 which mean you get 4 power points for free that you can buy adept powers with. This only apply to adepts (not mystic adepts, magicians, aspected magicians or technomancers).
For example Combat Sense 0.5 points + Improved Reflexes II 2.0 points + Improved Ability (Con) 0.5 points + Rapid Healing 0.5 points + Vocal Control 0.5 points = 4.0 points.
After game start, I spend 25 Karma and increase Magic to 5. Now i can spend next 5 Karma for a new spell, when im Magician.
After game start as a full magicain you can train any number of spells for 5 karma each.
Your current magic rating is not part of the equation. There is no limitation here like it is for power points.
You don't need to raise your magic to 5 in order to buy more spells as a magician.
(but since Magic rating is part of the spellcasting test you probably want to have a high magic rating anyway).
This also applies to aspected magicians and mystic adepts, but not adepts or technomancers.
As adept i spend 25 Karma increase Magic to 5 and as rule say my Power points are 5 too.
You get one extra power point because you increased your magic rating, yes.
...but there are a lot of cases your number of power points will not be the same as your current magic rating.
You could for example instead just spend 11 karma to initiate and when you initiate you get to pick a metamagic.
As adept you can choose to pick the metamagic "Power Point (Adept Only)" rather than another metamagic.
This way you would end up still having a current magic rating of 4 but now you would have 5 power points.
This only apply for adepts (not mystic adepts, full magicians, aspected magicians or technomancers).
Now a can add next Adept power to remained+new PP freely or like spell i have to spend some Karma for a new Adept power?
It does not cost karma to convert a power point into adept powers once you already have the power point.
(It is sometimes a bit hard to follow what you are actually typing. Have you considered using built in spell check of the browser and set it to English? I am also not native English speaking and I find it help me a lot when I am unsure about spelling).
