Before you can apply for a medical residency, you first need to obtain a Swedish medical license.
There are two paths to getting a Swedish medical license: you either already have a medical license in a country which Sweden has a mutual recognition agreement with (such as every European Union country) or you go through the Swedish medical licensing process with "general service" followed by the licensing exam.
Since the Portuguese medical degree programs do not award medical licenses at graduation (to the best of my knowledge, please correct me if I am wrong), your path forward in Sweden would be the Swedish medical licensing process.
You could also obtain a Portuguese medical license and have it recognised here but for the sake of learning the language and getting familiar with the Swedish ways of working I would recommend going the Swedish route.
The Swedish medical licensing process starts with up to 2 years of paid internship, "AT" (acronym for "allmäntjänstgöring", translatable as "general service"). After completing the internship period you are allowed to take the licensing exam, "the AT test", which is considered by most people to be pretty easy. After getting your Swedish medical license you can start applying for residencies.
The AT periods are applied to specifically rather than through a national admission process. Essentially, it is a job application rather than an education application.
The AT period is usually 18 months long but there are also a number of AT blocks consisting of 21 months or more. There are also a few special AT blocks of even greater length, for those who wish to pursue research during their AT period.
The average time between graduation and the beginning of an AT period is 9 months and growing (it is commonly believed that the county councils have failed to keep up the pace with the growing number of medical students) and last year it took 17 months, on average, to get an AT position in Stockholm.
Interestingly enough, the average delay times for the 21-month blocks are more than 3 months less than those for the 18-month blocks... meaning that if you are prepared to do a 21-month block, you are likely to finish before someone who is only prepared to do 18-month blocks.
The more peripheral areas (especially the north, the midwest and the southeast) also have significantly lower average times, however, and their payment is usually also better (the starting salary could be up to 30.000 SEK before taxes, rather than the 25.000 SEK which you can expect in highly competitive areas like Stockholm and Scania / Skåne).
Of course, having a great CV or personal connections would also affect how quickly and easily you get into an internship in a more competitive area. Many Swedish medical graduates work as "underdoctors" for many months in order to gain connections and improve their CV for the application to an internship area they strongly prefer.
Swedish language skills are not formally required for EU medical graduates (as far as I know) but following your licensing you will be expected by the authorities to have sufficient proficiency to communicate satisfactorily with patients.
(Most Swedes know enough English for basic communication - most of the imported movies and television series use subtitles rather than dubbing - but it is desirable that a doctor can understand the patients in their own language.)
You can find the internship application form and information here:
http://www.socialstyrelsen.se/applicati ... internship
For learning Swedish, the following web page is supposedly recommended:
I hope this information was helpful, feel free to ask more questions if anything is unclear or if you think of anything else!