First acquire the apothecary skill. Then go to the vendor and purchase 20 used vials, 20 musty elvish parsley, and 30 cloudy waters. Open up the Apothecary screen. Place the used vials in the container slot and the musty elvish parsely in the main ingredient slot. Before proceeding note the meter on the right part of the apothecary screen, it is the stability or stabilizer meter. The more filled this meter is, the higher the stability of the brew. Now place three cloudy water in this slot. The stability meter should be filled with a note on the screen saying, "This potion will definitely succeed," (or something to that affect) and then click the brew button. Volia! A potion has been brewed. Keep doing this until the apothecary skill has been leveled to about 20 to 25.
A couple notes at this point. First, one of the brews will probably be a critical success meaning that the potion made will be above the player's current skill level. Second, the player should have a stack of restoration potions which the player should keep. Even though at this point the potions are not that strong it is still good to keep a stack of restoration potions nearby, just in case.
Lastly, at this point in the apothecary leveling process, make sure to start building pipelines for ingredients that are from other gathering professions. For example if the player is a butcher start making connections with other players that are scavengers and cultivators. Each gathering profession provides certain ingredients that are necessary for apothecaries. Butchering and cultivating provide main ingredients and scavenging is the only gathering source for stabilizers. Or, if other players aren't available, either randomly or from the player's guild, the auction house more than likely will have the material the player is looking for.
Here is a few short hints and tips that will help the player and their attain goals as a apothecary:
Always carry containers, main ingredients, and stabilizers. Potions of all varities are very handy in all situations that might be encountered.
Stabilizers are an apothecary's best friend. Stabilizers, more often than not, will determine whether the potion that is being brewed will successful.
Getting to the higher levels of the apothecary profession requires the highest level of main ingredient that the player's skill allows.
Always check the auction house for ingredients that might be missing in the player's stock. It might be a little cost inflated but it is worth getting to the next tier of apothecary production.
Finally, familiarize yourself with what the main ingredients do in terms of adding effects to the final potion. Here is a link that provides a detailed list of what ingredient does what and what level the apothecary skill needs to be in order make use of that ingredient.