Water is present in all living trees as sap. Depending on the species there can be more water in wood by weight than there is wood fibre. Moisture content in wood is expressed as the percentage of water by weight compared to the weight of oven dried wood.
If the weight of water in a piece of wood is the same as the oven dried weight of the wood then the moisture content is 100%. For wood to burn well it needs to be less than 25%, preferably between 12% and 20%. Except in very dry climates 12% is as low as wood moisture content will get.
The best way to tell if wood is dry enough to burn is to test it with an electrical resistance moisture meter. FAA members will have one of these. Otherwise you can usually pick dry wood from green by its weight, it will feel comparatively heavy. Or you can split a piece in half and see if the middle is a lot darker than the outside.
The surface of freshly split wood will also feel cool to the touch if it is green due to the water evaporating. As a rough rule of thumb it will take about 6-12 months for green wood to get below 25% if it has been cross-cut into 300 mm rounds (this depends on the diameter of the log and the initial moisture content).
The answer depends to a large extent on what wood is available. For example in Western Australia, Jarrah and Wandoo are considered the best. In Tasmania, Brown Peppermint is considered best. In South Australia, Victoria and southern NSW it is generally River Red Gum. In Queensland, Ironbark and Box are preferred.
It also depends on what you are using the wood for. Red Gum is excellent in a slow combustion heater but does not burn with a lot of flame, so other species are usually preferred for open fires. Some species are known to not burn well at all, Turpentine and White Stringybark being two of these.
Each species has its own characteristics of burning rate, flame, coal and ash generation, which mainly relate to wood density and the chemical composition of tannins etc.
Perhaps the best thing to do is to try a range of the available species and pick the most suitable, which may be a mix of quicker and slower burning species.