Reading the plays can be heavy going if you're unused to the language.
Think of it like old poetry.
Not every native speaker gets it - some of my friends don't, but with the exception of one or two for whom it really does sound like a foreign language, most of them can still enjoy it. Don't stress about understanding all of it first time, there might be sentences which go over your head but through more exposure you'll pick them up.
What edition(s) are you using? It can be really helpful to read the plays alongside a school study guide if you're new to it, like Sparknotes (do they still have that? Used to be able to access for free online). Or there are editions with notes and which will offer summaries and explains words/phrases.
Remember Shakespeare mostly wrote in iambic pentameter too, which is the rhythm of 5 syllables (see here: https://www.studiobinder.com/blog/what-is-iambic-pentameter-definition-literature/)
One of the ways this is brilliant in his plays is because if you were to recite the lines according to the pentameter, the emotion and emphasis naturally falls where it should.
As well as watching the plays, following them along with the text can really help, as can looking up the plot to understand who's who and what happens before you get started.
Go into your local Waterstones - or even email the Globe or the RSC shop - to ask for advice on editions of the plays with good notes, but there is also loads online explaining the language simply.
Once you start picking it up, it'll get much easier and gcse study guides are really helpful if you're going into it new. A good guide will also explain cultural references that we don't have/use any more (like classical references or the use of 'Sir' vs 'Sirrah'...'Sir' was an address of respect, 'Sirrah' was used to talk down to someone of a lower class - Shakespeare often uses it in humorous lines or to indicate status). The BBC's GCSE Bitesize website might have some good guides to some of the plays, if they do that will be free to access.
For watching the plays, there are usually some available free on streaming (eg iplayer), or cheaply on Prime, or you can watch via the BFI player, Globe player or National Theatre. Your library might have some of the older ones too, like with Laurence Olivier. Baz Luhrmann's R+J was always brilliant for bringing Romeo and Juliet to life - it's pretty faithful to the text and makes it easy to understand what's going on. Great film actually, I need to rewatch.
Like others have mentioned, the RSC's 2008 Hamlet with David Tennant and Patrick Stewart was brilliant but Andrew Scott more recently gave the best To Be Or Not To Be soliloquy that I've ever seen. And yes, the Original Pronunciation performances are excellent.
I've a soft spot for the 1999 Midsummer Night's Dream, Hollywood cast but very good - Kevin Kline was excellent as Bottom.
Do you live near London? If you get the opportunity to see Shakespeare performed at The Globe you must go!
Don't forget the Sonnets too, they're short and beautiful.
It really doesn't matter what form you enjoy Shakespeare in, you might not ever get into reading the plays and that's fine, they were meant to be performed not read. But you also might find that you prefer watching performances which aren't such sticklers to the text too, that's fine as well :)