I can't help you with the wood/charcoal question for a Brinkman--I use a MES 30 (electric) and an AMNPS for smoke. However, I hope this will help you get a handle on the different types of wood.
I would also suggest that if you haven't already done so, sign up for Jeff's 5 day introduction to smoking.
I copied this from one of Dutch's posts:
This is information that was provided as a download from bigdaddyviking. . .
Reference guide for Woods used to Smoke Food
ACACIA - these trees are in the same family as mesquite. When burned in a smoker, acacia has a flavor similar to mesquite but not quite as heavy. Is a very hot burning wood.
ALDER - Very delicate with a hint of sweetness. Good with fish, pork, poultry, and light-meat game birds.
ALMOND - A sweet smoke flavor, light ash. Good with all meats.
APPLE - Very mild with a subtle fruity flavor, slightly sweet. Good with poultry (turns skin dark brown) and pork.
ASH - Fast burner, light but distinctive flavor. Good with fish and red meats.
BIRCH - Medium-hard wood with a flavor similar to maple. Good with pork and poultry.
CHERRY - Mild and fruity. Good with poultry, pork and beef. Some List members say the cherry wood is the best wood for smoking. Wood from chokecherry trees may produce a bitter flavor.
COTTONWOOD - It is a softer wood than alder and very subtle in flavor. Use it for fuel but use some chunks of other woods (hickory, oak, pecan) for more flavor. Don't use green cottonwood for smoking.
CRABAPPLE - Similar to apple wood.
GRAPEVINES - Tart. Provides a lot of smoke. Rich and fruity. Good with poultry, red meats, game and lamb.
HICKORY - Most commonly used wood for smoking--the King of smoking woods. Sweet to strong, heavy bacon flavor. Good with pork, ham and beef.
LILAC - Very light, subtle with a hint of floral. Good with seafood and lamb.
MAPLE - Smoky, mellow and slightly sweet. Good with pork, poultry, cheese, and small game birds.
MESQUITE - Strong earthy flavor. Good with beef, fish, chicken, and game. One of the hottest burning woods.
MULBERRY - The smell is sweet and reminds one of apple.
OAK - Heavy smoke flavor--the Queen of smoking wood. RED OAK is good on ribs, WHITE OAK makes the best coals for longer burning. All oak varieties reported as suitable for smoking. Good with red meat, pork, fish and heavy game.
ORANGE, LEMON and GRAPEFRUIT - Produces a nice mild smoky flavor. Excellent with beef, pork, fish and poultry.
PEAR - A nice subtle smoke flavor. Much like apple. Excellent with chicken and pork.
PECAN - Sweet and mild with a flavor similar to hickory. Tasty with a subtle character. Good with poultry, beef, pork and cheese. Pecan is an all-around superior smoking wood.
SWEET FRUIT WOODS - APRICOT, PLUM, PEACH, NECTARINE - Great on most white or pink meats, including chicken, turkey, pork and fish. The flavor is milder and sweeter than hickory.
WALNUT - ENGLISH and BLACK - Very heavy smoke flavor, usually mixed with lighter woods like almond, pear or apple. Can be bitter if used alone. Good with red meats and game.
Other internet sources report that wood from the following trees is suitable for smoking: AVOCADO, BAY, CARROTWOOD, KIAWE, MADRONE, MANZANITA, GUAVA, OLIVE, BEECH, BUTTERNUT, FIG, GUM, CHESTNUT, HACKBERRY, PIMIENTO, PERSIMMON, and WILLOW. The ornamental varieties of fruit trees (i.e. pear, cherry, apple, etc.) are also suitable for smoking.
Types of wood that is unsuitable or even poisonous when used for grilling. Don't use any wood from conifer trees, such as PINE, FIR, SPRUCE, REDWOOD, CEDAR, CYPRESS, etc.
There are many trees and shrubs in this world that contain chemicals toxic to humans--toxins that can even survive the burning process. Remember, you are going to eat the meat that you grill and the smoke particles and chemicals from the wood and what may be on or in the wood are going to get on and in the meat. Use only wood for grilling that you are sure of.
If you have some wood and do not know what it is, DO NOT USE IT FOR GRILLING FOOD. Burn it in your fireplace but not your smoker.
Also ELM and EUCALYPTUS wood is unsuitable for smoking, as is the wood from SASSAFRAS, SYCAMORE and LIQUID AMBER trees.
Here are some more woods that you should not to use for smoking:
Never use lumber scraps, either new or used. First, you cannot know for sure what kind of wood it is; second, the wood may have been chemically treated; third, you have no idea where the wood may have been or how it was used. For all you know, that free oak planking could have been used in a sewage treatment plant.
Never use any wood that has been painted or stained. Paint and stains can impart a bitter taste to the meat and old paint often contains lead.
Do not use wood scraps from a furniture manufacturer as this wood is often chemically treated.
Never use wood from old pallets. Many pallets are treated with chemicals that can be hazardous to your health and the pallet may have been used to carry chemicals or poison.
Avoid old wood that is covered with mold and fungus that can impart a bad taste to your meat.
Hope this helps a bit. FYI, not all woods were mentioned (my favorite, hickory for one).
The search bar at the top is an unbelievable fount of useful information--just search for almost anything and you'll find it.
EDIT: Just reread the list, and yep, hickory IS there.