We use always to say that something happens all the time (or very often).

I always go swimming on Monday evenings.
She has always wanted to visit Finland.

We use always to say that something happens all the time (in an annoying way).

He is always forgetting things.
He's always complaining.

We use always to say that something exists for ever.

I'll always remember the first time I met her.

Always is usually before the main verb;
after an auxiliary or modal verb;
after the verb 'to be'.

I always have...
I can always...
He is always...

Still means 'continuing later than expected.'
It expresses that someting started in the past and is the same now.

She is still living with her parents.
Why are you still talking about it?
They still haven't mended my car (They haven't mended my car yet).

Still usually comes before the main verb.
Still usually comes after the main verb if the main verb is 'be'.

I still like her.
Does he still live in Cambridge?
He is still here.
I can still run 100 metres in less than 12 seconds.