To get a divorce you must file a divorce application with the court. This can be done jointly or alone. A joint application for divorce is a request that is made by both spouses for a divorce order, with or without other terms (for example agreed upon child support payments). With a joint application, both spouses must complete the documents that are necessary in order to obtain the divorce and it can proceed through the court process more easily.
If you file a divorce application alone, your spouse must be served with court papers and given a 30 day opportunity to respond to your materials. If your spouse does not have a good reason why you cannot be divorced, or simply does not respond to your papers within 30 days, your divorce can go ahead.
In terms of the financial costs of a divorce, typically the person who is applying for the divorce must bear the full cost. If you wish to apply for a divorce, you must pay your lawyer and the court fees and there is little to no opportunity to recover these from your spouse. You could attempt to come to an agreement regarding fee sharing with your spouse before you commence your application, but if you want it more than they do, you will have little leverage with which to convince them to contribute.
If you also need to take your spouse to court regarding other issues such as support or custody you can sometimes recover part of your legal fees in the form of a costs award if they have acted in an unreasonable manner which has caused you to incur additional expenses.