As described in the Wikipedia's article on Super Mario 64, Bob-omb Battlefield is "the first course in the game, is a bright grassland, following the tradition of earlier Mario games. Here, Mario meets the friendly pink "Bob-omb Buddies," who are at war with the black Bob-ombs. The black Bob-ombs are led by King Bob-omb, who waits at the summit of a mountain as a boss character, guarding the first star in the game. In this course, Mario also meets a Chain Chomp and races Koopa the Quick, a Koopa Troopa with sprinting skills."
Star 1 - Big Bob-omb On The Summit
To get this star, you have to climb to the peak of hill and battle King Bob-Omb. To defeat this boss, run around behind him, grab him, pick him up, and throw him three times. After that, he will give you a star.
Star 2 - Footrace With Koopa The Quick
As you start the level, there will be a koopa near the starting point of the level. Talk to him and he will challenge you to a race to the top of the hill. Beat him and he will give you a star. Using the cannons is considered cheating.
Star 3 - Shoot To The Island In The Sky
To get this star, you must first speak to the pink bob-omb and get him to prepare the cannons for you. Having done that, all you need to do is shoot yourself to that island in the sky. Once you're there, just hit the yellow exclamation point box, and a star will pop out.
Star 4 - Find The 8 Red Coins
Find the 8 red coins and a star will appear somewhere on the field.
Star 5 - Wing To The Sky
To get this star, you must first shoot yourself, via the cannon, up to the island in the sky. If you've completed the Tower of the Wing Cap, there will be a red exclamation point box with a wing cap inside. Grab the cap and hop into the cannon on the island. Aim for the rings of coins, and if you get the center coin from each of the five rings, a star will appear on the island.
Star 6 - Behind Chain Chomp's Gate
From the start of the level, go over the bridge and run towards the large chain chomp. Ground stomp on the pole holding the chomp there four times, and the chomp will runaway after smashing the gate in front of the star. Just walk over and jump at the star to obtain it.
Star 7 - Get 100 Coins
As with every level in the game, getting 100 coins automatically nets you a star.