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Before I start, I took a look at other answers people wrote, but it still did not help me, as I can't understand.

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I tried finding the period of each function using [period/B], but what do I do next?

I can see its period is $2\pi + \pi + 2\pi/3$, and what do I do with that now?
I have periods of these separate functions, how do I combine them?

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    No, the period, if it exists, is the smallest among the (positive) common multiples of the period, here, it is $6 \pi.$ – 2017-02-03
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    Are you sure it is not simply $2\pi$? – 2017-02-03
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    typed into wolfram alpha and it gave me 2pi – 2017-02-03

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You have to take the lcm of the periods of the functions you are adding up. First one hase period $2\pi$, second one has period $\pi$ and third one has $2/3\pi$. Hence the period is $lcm(2\pi,\pi,2/3\pi)=2\pi$.

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    Thank you, so I just find LCM and get the period of the function. – 2017-02-03
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    You can't have lcm of irrational numbers. – 2017-02-04
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    The solution is not so easy as you imply. – 2017-02-04
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Hint: Take the largest period. Because the largest period will be important for the repetition of the signal in this case.

Note that: $x = z + 2\pi$ will give the same result. In general, you will have to determine the smallest common multiple of all the periods.

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    so I usually just take the largest period of one of the functions if it's multiple functions together? – 2017-02-03
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    No, not the largest period. If one period is 1 and the other 1.5, the common period is the least common multiple, i.e. in my case 3. For your case see the answer by Hyperplane. – 2017-02-03
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    @ BloodDrunk: In this case, it is the largest period. In general, you have to find the smallest common multiple of them as JeanMarie said in his comment. – 2017-02-03
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The key is to notice that if a function is, say, $\pi$-periodic, then it is also $n\pi$-periodic for all integers $n>1$. Then what you need is to find the smallest possible number $T$ such that $T = n_1 T_1 = n_2 T_2 = \ldots$ where the $T_i$ are the individual periods of your signals and $n_i$ are positive integers.

In your case $T_1 = 2\pi$, $T_2 = \pi$ and $T_3 = \frac{2}{3}\pi$, hence $T=2\pi$ with $n_1=1, n_2 = 2, n_3 = 3$