What do you want to 'get' from a gallery visit? Or think you should get?
What kind of galleries/art are you looking at? Permanent collections in traditional galleries (eg National Gallery), temporary exhibitions, commercial galleries?
Online resources: the gallery itself will probably have information on it's website. If it's a temporary exhibition there will probably be a large-print guide available - it's intended for visitors with limited vision but it's actually useful to read it to get the gist of what's in the exhibition before traipsing round and standing in front of the display boards. Sometimes you find yourself spending more time reading the labels than looking at the paintings - so might as well find somewhere to sit and read the large-print guide instead. Large galleries may well have the large-print guide on their website, probably in the accessibility section, so you can read it before/after your visit too.
In terms of thinking about what's in front of you, there's a broad variety of aspects you could consider but most of which you'd need to learn about outside the gallery.
Subject matter is an obvious one: figurative, portrait, still life, landscape, religious, historical, abstract, style of painting etc. Each could lead you down a path of discovery.
The context in which the painting was made, eg most religious art was produced for devotional purposes (on church altars or private worship at home) and, at the time, wouldn't have been considered 'art' in the way we think of it today, ie hanging on a wall to admire.
And of course the subject matter, so for Christian art recognising the life story of Jesus (eg Annunciation, nativity, crucifixion), biblical stories (woman with a man's severed head might be Judith and Holofernes, or Salome and John the Baptist), and the saints (a woman holding two eyes on a plate will be St Lucy).
There's so much to explore, most of which would mean reading around the subjects. I suggest online or library books, or you could find yourself amassing a personal library of art books (ask me how I know ...). You could start with the history of art, for an overview of the different genres and how they've changed over the centuries. Look at the websites for the big galleries such as the National Gallery and the Tate, as they have a lot of information and educational resources, even if you don't visit those particular galleries.
Ultimately, the thing to 'get' from a gallery visit is to see the works in real life, which will be a completely different experience to seeing reproduced images. Any 'learning' you want to do will mostly need to be done outside the gallery visit.