Here are a couple lessons that have helped me! Don't worry I'm not a spamming bot. I really love the TTMIK lessons <3

Sino-Korean - http://www.talktomeinkorean.com/lessons/l1l15/
Native-Korean - http://www.talktomeinkorean.com/lessons/l1l20/

This is just the beginning. Those two weeks of lessons? Plan to devote time not only memorizing the systems and usages but also counting units like 개, 명, 마리, 그루, 대, 채, 단, 과, 권, 켤레, 살, 송이, 통, 벌, 장, 병, 곡, 잔, 분, these are just a few to get you started... not to worry.. a beginner can rely on the first three (개, 명, 마리) for most situations. :)

Some time needs to be spent on learning to count with the first few numbers because some letters are dropped.. for example it is not 둘 개.. but 두 개.

I don't know which system is used for computation .. I can guess it is Sino-Korean.. but it is only a guess.

I probably left something out.. but this is all I could think of for now.
Good luck!

Ok... here we go.  Seat belts are not necessary, but pay attention to the road!

Uses of Pure Korean Numbers (하나, 둘, 셋..)

1. ages of people (except with older ages sometimes people use Sino-Korean.  I myself get confused with numbers above 50 so I often rely on Sino-Korean)

2. counting objects.. again for large numbers above 99 revert to Sino-Korean (.. sometimes I hear people using Sino-Korean in reference to numbers of objects like 40 or 50...etc)

3. hours in time both telling time and duration (unless the duration is in the higher range)

4. ordinal numbers (다섯째,...)

5. duration of months if using the counter 달 (한 달, 두 달..)

Uses of Sino-Korean Numbers (일, 이, 삼...)

1. years (both in number and duration 2014년, 이십 년 동안, ...)

2. months of the year 

3. days of the month

4. duration of months if using the counter 개월 (삼 개월)

4. minutes and seconds (time both telling time and duration)

5. money..won, dollars, cents, etc..

6. percentages, fractions, decimals

7. phone numbers

8. in front of words like 호 and 회, for addresses etc

9. measurements ..length, area, weight, etc.. 

9. numbers above 99

Part I... I need more room!

The simple answer is no, you don't have to learn both sets of numbers.....unless... you want to speak Korean.  :P

Yes, both sets of numbers are important and are required for specific situations.  It isn't easy and will certainly be a mental challenge not just to merely memorize both, but to be able to know which to use when!  A mental challenge is a good thing.  

화이팅!  ^^

Thank you so much!!! :D

<3