Here are some general interest mathematics journals that I like to read and perhaps to submit some of my works (for a list with many journals, see AMS link; there is also another list with rating: Aust MS list).
Notices of the American Mathematical Society is always readable with informative articles constituting good starting points to look for research topics of interest. The SIAM Review has similar features. The Mathematical Intelligencer is another good reading from the Springer.
The American Mathematical Monthly is a journal published by the MAA and it is considered to be one of the best of its kind. Mathematics Magazine is another journal from the MAA, so is The College Mathematics Journal and some of the articles that I read from these latter journals are quite interesting.
Mathematical Spectrum is a magazine for students and teachers of mathematics in schools, colleges and universities and for anyone interested in mathematics. Another similar magazine is The Mathematical Gazette which publishes articles about teaching and learning of mathematics. It is published by the Cambridge University Press.
Crux Mathematicorum is published by the Canadian Mathematical Society and it publishes articles and problems related math Olympiads and PME Journal is mainly by and for undergrads studying in math. Kvant Magazine is the Russian classic and it has been the first door to serious math and science for many people, including myself. Recreational Mathematics Magazine is another open access journal which publishes original and scientifically profound mathematical nuggets.
Arnold Mathematical Journal is a new journal published by the Springer. It aims to follow the footsteps of Dima (Vladimir) Arnold. Open Mathematics is an OA journal from De Gruyter.
The Fibonacci Quarterly is the main journal of the Fibonacci Association and it is an open access journal in number theory. The other similar journals are Hardy-Ramanujan Journal from the Episciences Project, Notes on Number Theory and Discrete Mathematics from Bulgaria and Moscow Journal of Combinatorics and Number Theory is from Moscow.
Forum Geometricorum publishes articles in elementary geometry which is a source of hundreds of challenging problems, it is open access. Elemente der Mathematik is a journal of the Swiss Mathematical Society and published by the EMS. Matematicki Vesnik is a journal of the Mathematical Society of Serbia and it is open access. Mathematical Communications is another OA journal from Croatia.
Documenta Mathematica is another open access journal, from Germany. Archiv der Mathematik is a Springer journal for short articles.
Journal of Combinatorics and Mathematical Research Letters are published by International Press with Chinese chief editors.
Theory of Computing is an OA journal on TCS.
Mathematical Medley is a journal of Singapore Mathematical Society. National Science Review is published on the behalf of the Chinese Academy of Sciences by the OUP. It is open access.
The Discussiones Mathematicae Graph Theory, Dependence Modeling, Tbilisi Mathematical Journal, Journal of Official Statistics, The Prague Bulletin of Mathematical Linguistics are all open access journals with no article processing fee from the De Gruyter Open.
Applicable Analysis and Discrete Mathematics is another OA journal from Serbia.
EMS Newsletter is another valuable open access newsletter. Other similar publications are Optima,Bernoulli News, Fields Notes, Asia Pacific Mathematics Newsletter and Pi in the Sky.