How to properly list quotes in a sentence

I suggest putting each slogan on a separate line of a block quote, thereby allowing you to remove the quotation marks. (Alternatively, use bullet points—also without quotation marks.)

The X company has three slogans:

Give people the power to build community and bring the world closer together.
Connect with friends and the world around you on Facebook.
Bringing the world closer together.

(I can't help but notice the lack of parallelism. The third slogan has a different verb tense than the other two. Somebody should have pointed out the inconsistency. Also, I suspect the first slogan should use a community or communities.)

If a vertical presentation is not allowed, then you either have to put the commas inside the quotation marks—or keep them outside but switch to single quotation marks. (That style of punctuation is allowed in scientific texts and literary criticism where literalism is required.)

Alternatively, and probably best, remove the quotation marks altogether and put each slogan—but not the commas—in italics.

In your comment, you seemed to be indicating that for your particular project you need to keep everything in one paragraph, rather than going to bullets.

I assume you're in the US, since you used double quotes, rather than single.

I know three ways of doing this.

As you had it, but with the commas and the period inside the quotes:

Three slogans are: "Give people the power to build community and bring the world closer together," "Connect with friends and the world around you on Facebook," and "Bringing the world closer together."

Similar to the above, but with semicolons:

Three slogans are: "Give people the power to build community and bring the world closer together;" "Connect with friends and the world around you on Facebook;" and "Bringing the world closer together."

With italics instead of quotation marks:

Three slogans are: Give people the power to build community and bring the world closer together, Connect with friends and the world around you on Facebook, and Bringing the world closer together.

Sometimes when things don't look aesthetically pleasing it can help to tweak the font, font size or line spacing a bit until it looks better.

Your questions:

My question is should I make it one big quote and place semicolons/ commas in between each quote?

No.

It looks strange to me having only the commas outside the quotations.

Look at how quotation marks and commas and periods are handled in some of your textbooks and you'll see that that's the norm.