Software development is one of those classic 80-20 activities -- ie where the first 80% of the work takes 20% of the time, and the last 20% of the work takes 80% of the time.
The key to speeding up your work in general is, therefore, to work on improving your performance for that that last 20% of the effort. This is typically the debugging stages and the work to tie everything together and make it usable.
You can improve this by taking more care in the early stages. Planning and design work at the outset of a project may seem like a waste of time when you're tempted to dive into the hardcore programming, but they can save a vast amount of time later on.
People will talk about specific methodologies like TDD or BDD. These are great, but if you're not doing them now they will take you a bit of time to adjust to, so they're probably not something that you want to be starting out with for a programming contest. But stopping to plan what you're doing before you dive in and do it will reap rewards no matter what. And they're the first steps toward TDD and BDD anyway. This is where you want to start.