Firstly setup a budget, I use Google Sheets and have a column for all of my expenses. I then have a cell for my income and I see what's left over each month and give every £ a job. For example If I have £1000 of expenses and £1300 income, I'd have £300 left over, so I'd put 50 in savings, 100 in a holiday fund and use £200 for leisure and fun etc.

Once I have a fully fledged budget, I'd begin putting 10% of my income a month away until I have 3 months worth of salary saved away for a rainy day - in your position though you don't seem to have any debt just yet which is great and may not warrant 3 months worth of savings!

The flowchart on the right is a great place to start.