Lux Classic Map
Classic Map or Lux Touch (free version)
If playing Lux Touch (free version) or the Classic map, the strategy is basically the same as with the board game. In one word: Australia. Conquering the continent of Australia is one of the easiest ways to secure victory.
Because Australia has only one access point it is easy to defend as you only need to defend that one territory. Once you have the continent, you already start receiving a minimum of 5 armies per turn and can use them to further your attack instead of worrying about defending multiple borders. Just remember to try and at least match the number of armies in this access point to the number of armies in neighboring enemy controlled territories and try to keep a minimum of 5 or so armies there. There are now 2 strategies that are commonly used, try to 1) gain another small continent, or 2) expand into Asia.
1) A popular strategy is to try to take another small continent, like South America or Africa. This means that you still need to have control of a territory inside of or close to one of those continents. Your target will depend on which one you are closer to or which continent has fewer opponent armies. Once you have a second continent, you will typically have a large advantage over the AI players and can proceed to take another continent. Just remember to fortify your borders well and quickly repel enemy attacks on those borders.
2) The other option is to expand into neighboring Asian territories. This can be easier to achieve than taking another continent at this point, but it is a slow way to go towards victory. It is very difficult to gain all of Asia and then hold onto it, but it's not necessarily the goal either. Remember that you also get extra armies for the number of territories you control, not just the continents. So even if you don't control all of Asia, but have a lot of the territories, you will also gain extra armies. Plus, you will have access to North America, Europe and Africa and can pick one of those to take over next.
But what if you didn't have any territories in or near Australia to start with? Then go for the 2nd easiest continent: South America. This continent is also easy to control as it only has 4 territories, but it is a bit more difficult to keep than Australia because it has 2 access points that need to be defended. Still, it's easier than the other options. Once you have South America, you should probably pick between North America or Africa as your next target. Don't try and go for both, just pick one and stick to it.
Of course, these strategies are more for beginner level players, if you are an advanced player, you might want to try and win by starting with a different part of the world, like North America or Europe.