Angels are certainly real; the Bible repeatedly tells us about them and the work God has given them to do. They are spiritual beings, the Bible says, who were created by God to do His will. The Bible says, "Are not all angels ministering spirits sent to serve those who will inherit salvation?" (Hebrews 1:14).
Because they are spiritual and not physical beings, the angels usually are invisible and unseen by us. In fact, one of heaven's joys, I believe, will be the joy of looking back over our lives and discovering just how often the angels intervened to save us or bless us -- although we weren't even aware of it at the time. The Psalmist said that God "will command his angels concerning you to guard you in all your ways" (Psalm 91:11).
Nevertheless, the Bible also tells us that at times the angels have become visible. When God announced to the Virgin Mary that she would bear His Son, He used the angel Gabriel to convey the news to her (see Luke 1:26-38). When Jesus' birth was announced to the shepherds outside Bethlehem, "a great company of the heavenly host appeared ... praising God" (Luke 2:13). Many other examples from the Bible could be cited.
Do angels still appear? I'm convinced they do on occasion -- although sometimes we may not even be aware of them, because they have chosen to appear as ordinary human beings. We should thank God for His angels, but at the same time, we shouldn't become overly preoccupied with them. Nor should we worship them, for only God -- Father, Son and Holy Spirit -- is worthy of our worship.