When using listing something using "and" it's almost always going to require are, but I was wondering if this sentence is correct. "Money and fame is not the answer." I question it since I feel like it acts as a group or set and is would be appropriate. Same with "Using kilograms and grams is a better measurement." Thanks in advance!
You've almost answered your own question - if two or more things are grouped together then you can sometimes refer to "them" as if they are one. However, it is not always so clear. It depends on whether you are referring to the collective as a single entity or a group.
For example, you might say:
There is a football team in my town.
"Football team" is the collective noun for a group of players, and you are referring to the team as a whole. However, if you were using the collective noun to refer to the players you would say:
The football team are walking onto the pitch.
This is because the collective noun is substituting for the individuals. It is basically the same as saying:
The players are walking onto the pitch.
When it comes to organisations the rules of grammar differ between British and US English. In BrEng we follow the same rule as above when speaking about organisations, for example:
The BBC is a broadcasting organisation in Great Britain
The BBC are currently filming in London.
Apparently, in AmEng organisations are always spoken of in the singular.
Referring to your example, I'm not completely sure if "money and fame" should be treated as one - they may come together, but not necessarily. However, in your sentence there is only one "answer", which would suggest that it has already been established "money and fame" are collectively "an answer" (singular), otherwise you would have to say "money and fame are not answers".
Consider these examples referring to "diet and exercise":
-"What do you think is the best health regime?"
-"Diet and exercise is the best regime for me".
This seems right because "diet and exercise" are being used collectively to describe the person's health regime.
-"What do you think is important to maintaining good health?"
-"Diet and exercise are very important".
This seems correct because diet and exercise are being cited as two individual factors.
Using the other example in your question, you could say "kilograms and grams are units of measurement" because they are both units individually; however they are also part of the same system of measurement, so you would say "kilograms and grams is a system of measurement".
Money and fame is not the answer - yes, this sounds ok, because "Money and fame" is an expression that can act as a singular concept. However,
Using kilograms and grams is a better measurement sounds wrong because the *using* of these units of measurement is not a *measurement* itself.