I dont really know much of either of them... I just want to start some sort of fund to grow my savings.
If you are dealing with a limited (1k-5k) amount of $$. And obviously you have limited investment knowledge .I would suggest you set up a Vanguard account for your kid and then direct deposit $50 to $100 every month into that fund . There are many funds but the most popular is the S&P500 Index Fund . Another option is if you know your time frame , say your saving for your daughters wedding then they have funds that start off more balanced towards stock and as the time of redemption arrives it shifts over to bonds to offset the risk of a sudden downturn when you are ready to withdraw. Its called Lifecycle or Target. You could easily do it on your own or call them for help setting it up.
Im not familiar with Sharebuilders and if they do this and if yes if they have any advantage.
If investing really scares you . I think there is a guy Ostreicher (Oakwood Financial?) who advertises a "Chasuna" plan in Lakewood ( hey!! thats where you are from). What he is doing is exactly what I described just he is putting it in a load fund . that means you are in essence paying him to help you. Which IMHO he rightfully deserves more then those few pennies if someone is too thick/lazy to figure out the ABC of squirreling away money in Mutual Funds for their kids