How do I learn to speak English really well?

Read everything and anything. Books, newspapers, magazines like the Economist, publications like the Harvard Business Review, and podcasts when you have time. Put Closed Captions on the TV so you are fully exposed to the whole script, visually and audibly. You could also join Toastmasters to perfect your speech and delivery. Practice makes perfect. Certain roles like sales require a lot of speech, so there is the chance to get good there. Other roles like quant work aren't going to give you a lot of practice speaking. You'll have different levels of proficiency speaking, writing, and giving speeches so you can fine tune as needed.

My father was an English major at U Penn, so growing up my writing and speech had to be perfect. I was frustrated growing up in an environment like that, but happy now that he was hard on me in regards to English skills. Attention to detail in English helped me have a high attention to detail when learning other languages.

Also, if you really want to work at your English, if you don't know a word, look it up. It's a lot easier these days to just type "word definition" in Google than to physically look it up in a dictionary (old school). And when you're studying for the GMAT or GRE verbal, your vocabulary will increase. Check out the Magoosh General words for GRE and GMAT - it tests your vocabulary very well. It is an Apple app for free.