You can find Bruce and Roo at the end of the first section of Stage 2, which begins with motorcycles and payphone booths.
Bruce is a clown dressed in blue, and Roo is the a Kangaroo with blue shorts.
You want to focus all of your attacks on Bruce while trying not to get pissed at Roo for bothering you while you're trying to rescue him. If you want to get him away without damaging him, grab him from the front so he faces you, and throw him over your shoulder (behind you). When you throw him this way (as oppossed to say, slamming him to the ground, for example), he'll always land on his feet and not take any damage. Then you can use the free time to pound up on Bruce.
If Bruce gets too close to Roo, he'll use his whip on Roo causing Roo to get more aggressive against you, so try to keep the two split up.
What I like to do is play through Stage 1, then when you're fighting the Stage 1 boss, Shiva, hold Attack (A by default) as soon as you land the finishing blow, causing him to scream, and continue holding Attack until the beginning of stage 2 fully loads up. Then, if you get a game over, you can continue and select Shiva as a playable character. After choosing Shiva, I like to spam the X button as my main attack, which is like a knuckle-punch type of attack that never goes into a combo, allowing you to indefinitely attack without knocking the opponent down until he is dead (where with normal attacks you do 1 or 2 regular hits, then a stronger hit, then some kind of double hit that always knocks enemies to the ground). I usually use this on Bruce to take him out quick, as the knuckle-punch does a good amount of damage.
Some tips for the game: On the "press start" screen, Hold UP, and A, then press start until the next screen loads. This allows you to choose Roo without needing to unlock him.
Also, highlight "Battle" and hold UP and A again, then press start on options (since pressing and holding up moves your cursor from Battle to Options). This cheat allows you to enter a stage select screen (so you could start on stage 4 if you wanted or whatever stage you want).
If you can fire up a second controller and "join" the game by pressing start anywhere in the menu or title screen, when you go into the options (either by the above cheat or not), point the cursor with player 1 onto the "Lives" option, and then with the 2nd controller hold UP, X, A, B and then (continue holding those buttons) with the first controller increase your lives count, up to a maximum of 9.