Everyone's path is different, but this is how I did it.
Really, most of it comes from learning, which you should be doing anyway. But rather than skipping over words I don't understand, I look them up. Contrary to what your yetzer hara tells you, it will take less than a minute of your time.
The easiest way to do this is to go to Google and search "define X." The answer will not always be the first link, and you should always check a couple of links. There can be several ways to define a word (or spell it in English!), so reading several links will give you a better sense of the word and see it used in several contexts.
Prioritze links from websites you recognize as legitimate (as opposed to a random person's blog). Normally definitions will come from sites like Chabad and Aish simply because they're aimed at (and created for) people learning to be frum. If a page from one of those sites pops up, start there and then read the others.
How do you read these several links without losing two hours of your day? Don't read the whole page. Use ctrl+F (command+F for my fellow Mac users) to locate the word you wanted to define. Then only read the appropriate part of the article/page. Don't let yourself get sucked into the larger article unless you really have the time to read it. If you don't have the time, but the topic interests you, bookmark it for later.
Really, it's that simple. The trick is remembering the word the next time, which will almost never happen. Just keep looking up the word until you remember it. Studies suggest you need to "touch" a word approximately 7 times in order to remember it. You can increase those "touches" by reading the information out loud in addition to read it silently. Reading several sources can also help you retain the information faster.
What's the real problem? Learning to pronounce the word correctly. I can't claim mastery of that yet, as I mispronounce words all the time. Here, listen to how other people say the word. You can even ask your friends to model it for you! Listening to shiurim will passively import many words into your brain without you even realizing it.
But speaking more generally: in the Jewish world, mispronunciation isn't so awful because there are lots of ways to pronounce words, so surely you're correct according to someone! (And sometimes, you can blame it on your native accent, such as a Southern accent or French accent.) For example, I can pronounce a word perfectly fine for an American Jewish audience (this week's example: "MEvushal"), but my new Israeli roommate will look at me like I'm sputtering nonsense. Reversing that, if I use a "correct" Israeli pronunciation of a Hebrew word (this week's example: "yeshiVAH"), Americans may make me repeat the word until they realize what I'm saying. Likewise, maybe I just can't pronounce the non-English sounds of Hebrew correctly. Some days are better than others, and some sounds are better than others. And eventually, you get used to the "boy am I an idiot" feeling and realize no one cares. Except for the people who have to feel superior to other people, and those aren't the people you want to hang out with anyway, right?
So what is the basic lesson here? Have the humility to realize the limits of your knowledge and the willingness to learn from others. One day (or even today), you can provide the same kind of help to someone else. There is certainly plenty I have left to learn, but I share the knowledge I do have. You can do the same to help someone else!