(a) Ich war gestern draußen, ging am Abend ins Kino.
(b) Gestern war ich draußen, ging am Abend ins Kino.
Is each of the two sentences correct?
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The first parts of both sentences are alright and do not affect the following.
Joining sentences like this (no conjunction or repetition of subject) is grammatically correct, though very rare. It is usually used half-poetically to describe activities in a ”streaming” way without relating them too much to each other or emphasising the actual content. For example, in your sentences, you could use this sentence structure to express that yesterday was very relaxing and you did not do much or that yesterday was very boring (using a totally different voice).
Also, such a construction is used when more than two verbs are joined to describe a series of activities:
Gestern war ich draußen, ging im Park spazieren, aß in einem Restaurant, schaute mir einen Film im Kino an, ging nach Hause.
As already stated, it is much more common to use a conjunction before the last part in all of the above cases. So this is what I would strongly recommend to use when in doubt:
Gestern war ich draußen und ging am Abend ins Kino.
Gestern war ich draußen, ging im Park spazieren, aß in einem Restaurant, schaute mir einen Film im Kino an und ging nach Hause.
I'm not sure if I would call either "correct". I myself would never use either sentence as is. I'm not entirely sure what you're trying to achieve, but joining events with a comma does work, if you have a true list (more than a pair; pairs are joined with a conjunction).
So I'd say
(a) Ich war gestern draußen und bin am Abend ins Kino gegangen.
(b) Gestern war ich draußen und am Abend im Kino.
(a) Ich war gestern draußen, bin am Abend ins Kino gegangen und war trotzdem vor 9 im Bett.
(b) Gestern war ich draußen, ging am Abend ins Kino und um 8:30 Uhr ins Bett.