Well, it basically relies on your field of interest. For different fields there are different journals in which papers concerning that topic are published. Then again, many different journals are published by the same publisher, see for example Elsevier. Wikipedia has a list of journals.
Usually, unless they are open access or your university has a contract with the publisher, you need to pay money to read a paper or journal. But your local math library probably has a lot of old (and new) journal articles to read from. arxiv.org should also be mentioned, as perhaps the most well known source for open access prepapers as well as Google Scholar which can help you to find a specific paper given the title and/or author. But also note that the quality of journals differ, going down to journals which will publish anything for money (see also).
Reading new papers nowadays is challenging anyway, because they are basically the frontier of science and therefore often really specialized. For example, I can hardly make sense of papers about vector bundles because I didn't really ever had anything to do with them and so hardly know what they are anyway. If you want to start with a topic, it is usually more advisable to read a structured book about it, which usually contains a lot of references to papers, if you want to delve into it.
On another note, it can be very educational to read the original papers of great mathematicans. For example, there is the Euler Archive but there exist paper collections of almost all great mathematicians.