You can put "also" before "think" or before "quit". Each usage has a different meaning. "I think that you should eat healthy food. I also think that you should quit drinking." (Here, "also" modifies "think". You think two things.) "I think that you should quit drinking and should also quit smoking." (Here, "also" modifies "quit". There are two things you think the person needs to quit. "Also" comes after the modal verb and before the main verb.) You can use "also" at the beginning of a sentence as well (followed by a comma), which has the same meaning as saying "furthermore" or "in addition". However, this is somewhat informal, and in formal writing it's better to use a more formal adverb to begin a sentence: "furthermore", etc.

The adverb wouldn't normally go before the modal verb there. Also, you can remove the 'that' for conciseness.

There are degrees of incorrectness. *"I think that also you should quit smoking" is completely, utterly, entirely wrong. *"I think that you also should quit smoking" sounds bad, but not nearly as bad as the first sentence. When Amy Lightfoot was making her list on the BBC link you sent, she wrote down everything a native might conceivably say, and called it all correct. A native MIGHT say that sentence, but they would be MUCH more likely to put the adverb after the modal. If they put the adverb in front of the modal on a written essay, a teacher would almost definitely mark it wrong. Here's a chart showing use of "you should also" and "you also should". As you can see, we DO occasionally use "you also should", but it is much, much less common than "you should also." Follow this link: goo.gl/nMk4WQ Also, you are "confused". Amy Lightfoot is confusing--you aren't.