Translation:My sister is sitting and reading a magazine.
I personally understand «Моя́ сестра́ сиди́т и чита́ет журна́л» literally: i.e. my sister is reading a magazine while sitting.
There is some structure like that, but it's colloquial and I wouldn’t use it with «и»: «моя сестра́ сиди́т[,] чита́ет журна́л» (it’s so colloquial that I don’t even know if I need a comma here!), but I don’t think it means the action is ongoing.
So I'm not completely sure what's the meaning of this construction, but looking through examples in the National Corpus it seems it doesn't have any additional meanings except continuity..? Not sure.
However, as I've said, it's not used in this very sentence: I think «и» breaks everything.
The verb forms would not be any different. Russian only has one present tense to describe ongoing processes as well as habitual actions.
The only place where it matters is verbs of motion. We use different verbs for one-way motion and for multidirectional/aimless motion—and habitual motion to a place almost always implies a trip back (I mean, in actual usage). On the other hand, if you are moving RIGHT NOW, the only meaning a multidirectional verb can have is moving around without any particular destination (like being on a walk or driving all around town). So "I go to the movies every week" and "I am going (I am on my way) to the movie theatre" will use different verbs.
Probably, only идти/ходить ("walk") and ехать/ездить ("go by vehicle") matter to a complete beginner. For a more advanced newbie a few more are also useful: бежать/бегать (run), плыть/плавать (move by water), лететь/летать (to fly), нести/носить (to carry/bring an object manually), везти/возить (to carry/bring/take by vehicle). There are 14 or 18 or these verbs pairs, depending on whether you include reflexive variants some of these verbs have.
Журнал looks the same in accusative and in nominative case. In fact, all the inanimate masculine nouns do.