I sympathize. Did you read them aloud at all? That is the one sure way to get the feel for Shakespeare. Wrap your voice around the lines and you will almost immediately have a better sense of what is going on. Will you understand every single line right away? Of course not, but remember that virtually every word in Shakespeare is still in the dictionary. He's writing not in Old English, but in early Modern English, and he is an artist who works in words.

Notice how frequently he summarizes in a less difficult line composed of one- or two-syllable words the more complex images and ideas that a character has been saying in a soliloquy. In fact, most of the most recognizable, frequently quoted lines in Shakespeare are composed of one- or two-syllable words that any child's vocabulary contains. (I give you "To be or not to be..." for example.)

Another hint: read the poetry as if it's prose, and the prose as if it's poetry. In other words, break yourself of the habit of reading a line of poetry, pausing Dr. Seuss-style, and then reading the next line. Shakespeare's lines spill over into the next line frequently (called Enjambment), and reading the planes of poetry accordingly make a huge difference. Likewise, (and this is why you must read aloud), take a breath when you're reading the prose passages to help you to hear the power of the images that Shakespeare creates.

Also, I would recommend choosing a play to read that you can also watch or listen to as you're reading. You can then stop and start  the spoken version if you need to reread or follow what's going on. Again, seeing and hearing how the lines are read, both for meaning and for feeling, can go a long way toward helping one's understanding of the lines. If you can, take in a theatrical performance of a play. Often, that can be an eye-opener as well.

You can usually find the BBC versions of the plays that were made in the late 70s and early 80s on-line. Most are essentially straightforward versions, but they are easy to follow with text in hand, as they are not "cinematic" in style. I would not shy away from the recent spate of Shakespeare films, either. It's just that the BBC versions were made with the purpose of having a performed version of the text of each play always available.

I would recommend using two or three annotated editions, too. If you're going paperbacks, the Folger Library editions (Washington Square Press) are old reliables that have been updated with excellent articles and background information, the groundwork that you may be referring to. I also recommend the Penguin versions, which are equally accessible. Both are inexpensive and easy to find. 

Don't know what you've read already, though most of us know Romeo and Juliet and/or Macbeth well from high school. Start with one like that. You know the stories and the characters. Those two are not particularly long and the language is not complicated, especially compared to some of the later plays. Reread a summary without worrying about spoiler alerts. Every one of the plays rewards rereading, trust me.Time enough for plunging into a new play once you get your (metric) feet wet in familiar waters. 

Please don't give up on this noble pursuit. Shakespeare is you often ruined for us by teachers who assign it to be "read" overnight, as something to be gotten through, without recognizing the absolute necessity of performing it in their classes. Others, whose distaste for Shakespeare, born of a lack of understanding, wind up coloring their students' attitudes toward it, making Shakespeare seem as if it's like a foul-tasting medicine that you have to take.

Stick with it, Trebor. It's hard at times, but it's the hard that makes it great, as Jimmy Dugan says.