Correct me if I'm wrong...

When you open a Roth and contribute money to the account, that money is used to invest in a fund (a mix of stocks, bonds, and other investments) of some kind.

Mutual funds usually have an minimum investment amount; i.e. you must have at least $1000 (for example) to invest in the fund.

At Fidelity, there are two options
If you opened your account with at least the minimum amount ($1000), you can immediately invest in the fund and your money starts growing.

If you opened your account by making automatic contributions on a monthly or quarterly basis (say, $100 a month), you can avoid the initial $1000 investment. But, the money just sits there until there is enough accumulated to make the initial investment of the fund of your choice.

Without having enough money in the account to make an initial investment in a fund, the money itself is not growing.

Right?

That said, should one open an account with a mutual fund company by making small monthly contributions and allowing the money to accumulate until you reach the minimum investment requirement? Or, should one just save money on their own (savings account, short-term CD, etc.) until they have enough to make the minimum account opening money, and then open the account?

I guess I don't understand the value of opening an account by automatic contributions if you can't afford to invest in any of the funds. For example, if the minimum investment for a fund is $1000, and you are contributing $100/month, it would take 10 months before you could invest in anything.

Your money is probably in a money fund earning some nominal rate of interest. It is better to establish the habit and get started putting the $ into the ROTH. If you are disciplined enough to save the $ outside of the ROTH you still lose the tax advantage on the smalll amount of earnings on the money market account that may be tax free when you withdraw it much later on.

In your case, build up your funds and when you hit the minimum transfer to a fund.

Many funds will allow you to start and IRA at a lower amount if you commit to a monthly automatic contribution. I don't know if Fidelity has this - so this section option might apply to others who have not yet picked their custodian.

Your alternative approach also works. The difference is so tiny (taxes owed) that its a non-factor. The bigger hurtle is to dedicate a significant amount of your current income to the plan. A $100 per month only puts $1,200 to work each year!

Note, purchasing a mutual fund in even $$ amounts each month is "dollar cost averaging"... which means you buy more shares when the fund is down, a slight bow to the "buy low" concept.