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The Menu button looks like three stacked dots. You can search by Skype name, profile name, email address, or phone number. It's best to search for other users by their Skype name or email address. If you search by a person's actual name, there will likely be multiple matching results. This will open a text chat with the person. Depending on the other person's privacy settings, you may not be able to send them messages until they've been added to your contacts list. The other person will need to approve before they are added to your list.
Tap the "Menu" button and then "Add people". Search for the person you want to add. Tap the person you want to add in the list of search results. Tap "Add to contacts" to add the person to your contacts list.