You are in luck, I have some knowledge in this field.

Firstly, will you be starting a new public company from scratch? That would be very expensive and costly. The cheapest route would be a reverse merger with an existing shell, costs are around $100k-$300k depending on the structure of the deal and the quality of the company shell.

Raising money privately would be an ideal situation if you have the investors. When you are public you are at the mercy of daytraders, shorters, flipflop investors, market makers and general market conditions.

BUT, when done correctly, you can probably raise a lot more money in the public markets than privately.

Be aware that you gotta have one helluva business plan and tons of potential to get anybody excited about your public co.