I'd use the same list that Christophe Strobbe referenced, but with a few corrections.
As he stated, the top 10 most spoken languages in the world (excluding English), ordered by native speakers, are Mandarin, Spanish, Hindi, Arabic, Portuguese, Bengali, Russian, Japanese, Punjabi, and German.
It isn't necessary to learn all of these languages to be able to communicate with their native speakers; keep in mind, this list is based on the number of native speakers only. Many people who speak one of these languages speak another as well, meaning it isn't necessary to learn all of these.
For one, after several hundred years of British colonialism in India and Bangladesh, the teaching of English in schools is emphasized, and thus most Indians or Bangladeshis born within the last 60 years are fully fluent in English, if not at least mostly literate. Because of this, Hindi, Punjabi, and Bengali really aren't vital to know; you can communicate with most, if not all, Hindi, Punjabi, and Bengali speakers with just English.
In a nutshell, the answer to the question What are the most common languages among people who DON'T speak English? is the first list I provided. However, if you want to make a better decision and choose langauages whose group of speakers will not overlap with each other, you should go with this better revised list:
Mandarin, Spanish, Arabic, Portuguese, Russian, Japanese, and German.
The easiest to learn out of this list for an English speaker would be Spanish, Portuguese, and German.