Whether you have a woodburner or a multi-fuel stove, it is important that any wood that you burn is well-seasoned (dry). Some woods take longer to season than others but, generally speaking, wood should be cut, split into logs and dried outside, but under cover, for a year before being used as fuel.
Wood from conifer trees, like pine, tends to contain quite a lot of resin and is good to use as kindling. If you know any cabinet makers or kitchen manufacturers that use pine, ask them if they have any off-cuts which they want to get rid of. You could find that you are never short of kindling again!
Both ash and beech wood are considered excellent for burning as they produce a steady flame and good heat output. Hawthorn is also good for burning in a woodburner and burns at a slower rate that beech or ash. Horse chestnut produces a good flame but you may find that you need to close the doors of your stove as it has a tendency to spit.