Here are the relevant rules for each one.
When a creature with this special attack makes a charge, it can make a full attack (including rake attacks if the creature also has the rake ability).
Format: pounce; Location: Special Attacks.
If a creature with this special attack hits with the indicated attack (usually a claw or bite attack), it deals normal damage and attempts to start a grapple as a free action without provoking an attack of opportunity. Unless otherwise noted, grab can only be used against targets of a size equal to or smaller than the creature with this ability. If the creature can use grab on creatures of other sizes, it is noted in the creature's Special Attacks line. The creature has the option to conduct the grapple normally, or simply use the part of its body it used in the grab to hold the opponent. If it chooses to do the latter, it takes a –20 penalty on its CMB check to make and maintain the grapple, but does not gain the grappled condition itself. A successful hold does not deal any extra damage unless the creature also has the constrict special attack. If the creature does not constrict, each successful grapple check it makes during successive rounds automatically deals the damage indicated for the attack that established the hold. Otherwise, it deals constriction damage as well (the amount is given in the creature’s descriptive text).
Creatures with the grab special attack receive a +4 bonus on combat maneuver checks made to start and maintain a grapple.
Format: grab; Location: individual attacks and special attacks.
Rake (Ex)
A creature with this special attack gains extra natural attacks under certain conditions, typically when it grapples its foe. In addition to the options available to all grapplers, a monster with the rake ability gains two free claw attacks that it can use only against a grappled foe. The bonus and damage caused by these attacks is included in the creature’s description. A monster with the rake ability must begin its turn already grappling to use its rake—it can’t begin a grapple and rake in the same turn.
Format: rake (2 claws +8, 1d4+2); Location: Special Attacks.
Well, I use a lion companion in a game, and they are great.
First, always charge to get the +2 bonus on attack. Unlike other players, because of Pounce you get all your attacks on the charge. That's awesome!
Second, grab gives you the +4 to your CMB to grapple the opponent without giving the enemy an attack of opportunity. This is awesome, because it really hinders the enemy in performing many attacks against your companions, they pretty much need to deal with the grapple. I almost always try to grapple an opponent with grab, unless you have some really strange reason not to do so.
Assuming it's a tough critter and you have help with damage, you should then try to pin the enemy on your next turn assuming you maintained the grapple. While the lion is doing this, do your rake attacks! Hahaha.
Now, some technical issues. First, you must have your lion have the "attack" trick. If you want the lion to attack weird stuff like zombies, skeletons and outsiders, you will need to teach it the attack trick twice. It's DC20 to teach it, but since this is usually done during downtime it's not a problem. To then get the lion to attack, it's a DC10 check. For almost any Druid over the level of 4-5, this is like an automatic success and is done as a swift action. There are lots of tricks, but you should check with your DM to determine which tricks are okay to learn. You should get really familiar with the Handle Animal skill rules.
Also be aware that some GMs really have a problem with you being in a wildshape while handling your animal companion. While there isn't anything in the rules about you must be in human form or use common speech to command your animal companion, some GMs really get annoyed and will make it harder on you to command your animal. Some things to try: 1) wild-shape into a cat form so that you can speak with your lion companion, since you can communicate with your companion.
2) Invest in the beast-speech feat. Total waste of a feat, but if your GM is a pain about commanding your companion, it might be required. Plus, it does add to role-playing capacity as a druid.
3) Increase your lion's intelligence to 3, so that it can understand common. Thus, it could understand commands from your friends. But, your GM might require your companions to do the Handle Animal check. It's DC10 but still, it might fail.
4) Just have your companion default to "guard" on you, which can be maintained pretty easily and it will defend you if you are attacked without a command.
5) Perhaps negotiate to have a -4 penalty for using Handle Animal while in wild-shape. This is totally a house-rule, but some GMs just really don't like the idea of you using Handle Animal skill while in a non-human-like shape. So, with wild-shape really kicking in around level 4, this shouldn't be too bad given the bonuses you receive as a druid.