How to Find a Game/Group
If you're looking for people to play casual (non-challenge) games with, start by checking the Looking for Group thread on the Unofficial BSN, since many regulars have offered their IDs for others to connect with them and play. Adding your information to that thread makes it simpler for the next person to find useful help.
When posting, include as much as possible of the following key information:
** Your Origin ID (on PC), OR your gamertag (on Xbox Live) OR your PSN ID OR your WiiU Mii tag. (Many people have different IDs on different platforms and/or on here on BSN.)
** Your timezone and the kind of times that you play. People from all round the world read BSN.
** Your preferences for difficulty level, Bronze/Silver/Gold/Platinum.
** Other preferences such as for a team with microphones, biotic/tech team, doing challenges, duos, etc.
Before searching for a game using in-game the match-making system, make sure that you have all the (FREE) multiplayer DLC installed:
Without these DLCs, you won't be able to play with anyone online.
(Note: if you happen to be on WiiU, ignore the DLC remarks. All relevant DLC was included with the edition of ME3 for the WiiU, and what wasn't has never been released for WiiU. Just put in the disc and you're good to go.)
If you can't find any games using in-game the match-making system - meaning that the searching goes on forever - please check that your router has UDP ports 3659, 5659, and 6000 open for inbound and outbound traffic. If you are on PC, you may need to open those ports for UDP in Windows Firewall as well. Even if you can get games without doing that, you will find that you can find games a lot faster once those three ports have been explicitly opened.
A tip: If you have the ports opened, and your search for a quick match still takes a long time, try going to the Store and loading each pack in turn (you don't need to buy anything), which resets the connection to the server. Loading the leaderboards can do the same thing. Then return to the search screen as soon as the Galaxy at War map has loaded.
Please note that there is apparently a version mismatch between different regions that prevents players from getting games because everyone needs to be using the same version of ME3. The details and the fix are described in a post on the ME3Tweaks site.
Don't be put off by being new. There are a lot of very able regular players who are perfectly happy to guide new players through the first few games (or challenges) on easier difficulty.