Welcome to the Forum 
Because you are from the EU, you have a right to stay in Sweden for 3 months and work. However, you will still need to get a Personnummer if you plan to stay in Sweden for the long run.
To apply, you will need to be able to either:
Have a job offer in Sweden from a Swedish company (and with proof – the easy way)
A Migrationsverket uppehållstillstånd (the right to stay in Sweden longer than 3/6 months and the card to prove it)
Evidence that you can support yourself with some form of regular income (with proof – really hard)
You will also need to show:
Your passport
Your European Health Insurance Card
Someone/something that can confirm your address
Ideally, a Swede to help with the process
Have all this ready in good time as the more you can bring, the more chance you have of your application being processed.
To Apply
You need to go in person to a Skatteverket Office () and there is no paperwork to fill in before – just bring ALL of your documents. Be as prepared as possible. Get there before 9am (there will be a long queue) and be brutal. Push through, run to the reception, tell them what you need and get the form to fill in. The Skatteverket Offices processes Personnummer and ID-Card (ID-Kort) applications. You just want to apply for a Personnummer!
Fill in the 4-page form they give you in as much detail as possible. If you have any questions then just ask at the desk.
You will then be passed to an advisor who will process your application.
The Application
The desks at the Skatteverket Offices are high up so you feel like an ant. Give your paperwork and start telling the advisor what they need. The more you have (and have documented), the better your Personnummer application will be.
The person you speak with will ask you questions regarding your application, what you are doing in Sweden, what you plan on doing here, etc.
After your Visit
The waiting times for a Personnummer are better than Migrationsverket and are something like:
1/2 weeks if you have Migrationsverket uppehållstillstånd
3/4 weeks if you are applying with some other grounds
Next Steps
Then it is a waiting game. I’ve heard stories of people getting their personummer in 1 week, and others can wait months. Unfortunately, the truth is that it really depends on the case worker you get.
You can read up on more from the in English.