After the league starts, your opponents can be found on the Matchups tab of the event page under the appropriate grouping for your UTR rating:
Your opponent's phone/email contact details are available on the match cards directly. If you go to the event page, Matchups tab while logged into your Universal Tennis account, you will see a phone icon next to your opponent's names. If you click on that, you'll see their preferred phone/email contact for reaching out to. :-)
Pro Tip: Be sure that you are logged into the Universal Tennis account linked to your league registration - if you are viewing through any other account the phone icons will not be visible, as they are only available to the players in each event draw.