해 봐야겠어요 is composed of two patterns Verb + 아/어/여 보다 : This pattern is used to: 1. Tell someone to try or attempt something and to see the result of doing so. 2. As a question this can be used to ask if someone has ever done or tried something, for example "미국에 가 봤어요?" could be translated to "Have you ever been to America?" 3. It can also be used to make a command sound less strong and demanding. ex) 이거 먹어 봐. Try this (food). 미국에 한번 가 봤어요. I've been to America once. 한번 마셔 보세요. Please try this one time (drink). 입어 보세요. Please try this on. Verb + 아/어 + 야겠다 : This pattern is used for when you think something is advisable, similar to "had better" or "(I) guess (I) should" or "(I) guess I have to." This pattern is often used in a way to reflect your own personal thought, so the regular 다 ending is often used (i.e. 가야겠다). This is less strong than the other patterns which stress obligation such as V + 아/어 + 야 되다, V + 아/어 + 야지 and V + 아/어 + 야 하다. ex) 나는 가야겠어. I guess I should go. 배고파. 밥 먹어야겠어. I'm hungry. I'll guess I should eat. 저는 이제 자야겠어요. I had better sleep (now). 택시를 타야겠어요. I guess we'll have to take a cab. Therefore, 해 봐야겠어요 means "(I) guess I have to try doing" 저도 내일부터 투이 씨를 따라 해 봐야겠어요. I also guess I have to try doing it as following Mr Tui from tomorrow.

-야겠- is the shortened form of -야 하겠- 봐야겠어요 > 봐야 하겠어요 V+아/어야 하다 means "have to + verb". For example 저는 가야 해요. - I have to go. 봐야 해요 = have to see Adding the particle -겠- expresses your will to do it. 봐야 하겠어요 > 봐야겠어요 = I'll have to see that BUT, in your sentence the 보다 does not mean "to see", it's used as an auxiliary to express that you're trying something new. 오늘 떡볶이 처음 먹어 봤어요. I tried tteokbokki for the first time today. So your sentence means something like "From tomorrow I'll (have to) try to do like Tui" Sorry my explanation is a bit messy, I hope you'll get it. ^-^