Golwg
Golwg is a lively magazine published weekly in colour, It contains articles on current affairs, feature, articles, articles on arts and sport. It is not specifically a magazine for learners but the style is generally quite readable and entertaining.
An offshoot is the Golwg360 website which is constantly updated with news and topical material.
Lingo
Lingo is published by the same organisation as Golwg and is aimed at learners. Articles are written with paragraphs at different levels of difficulty from beginner to intermediate, and it provides a good way of developing reading skills. You may well spot the occasional star from SSiW in it. It is published bi-monthly.
Y Cymro
The weekly Welsh newspaper. News about Wales, events, sport, art and poetry are all included. The style varies but is mostly rather formal, with no concessions to learners. Good practice later on when you have got used to Golwg.
Acen magazine is aimed at learners. It features a number of short articles suitable for beginners. Available for download at http://www.learnons4c.co.uk/magazine.php.
A Papur Bro (District Paper) is a community-based and community-produced Welsh language magazine, usually produced monthly. There are currently around 60 published across Wales. Many have content aimed at learners. A list of papurau bro can be found here.
Dan Y landsker was first published last March 1st with the support of Antur Teifi a business/project support group based in Newcastle Emlyn. I think it is available monthly.
http://www.bbc.co.uk/cymru/deorllewin/papurau_bro/dan_y_landsker/