"Do I need to show money or bank savings/proof of income to apply for a tourist visa?"
With the tourist visa, you need to show that you have the means to return to your home country. You also need to show a lot of things. You need to show that you have strong ties to your country. But, showing that you do have money in your bank account, you do have a savings account and you do have the means and resources to not only come to the United States and be a tourist, but more importantly, the money to return home is what's important. You want to show that you can afford the expense of coming to the United States.
If you are financially struggling, it will be assumed that you intend to be an economic refugee and plan to overstay a tourist visa in search of illegal work, which is an easy denial. To supplement your bank statements, another great strong tie is a stable income, from a job in your home country. If you are taking time off work, a letter from your employer authorizing time off for your vacation is another signal that you intend to return to your country.
The number one thing that will get you denied for the tourist visa, at least in our experience, is not being able to convince the adjudicator that you have no interest in staying in the United States. And one of the ways that you do that is show that you have a good life where you're at, you are making good money, you own property, you have kids who are not coming with you, and other things like that.
