Tilos is a lovely island not over-run with package tourists like the nearby islands of Symi and Halki. There aren't any really good sandy beaches but this does not bother some people.
There is a good circular walk around the SE corner of the island, this gives access to very steep tracks down to beaches on the south coast. There is a cliff path on the north side of the port around to Lethra beach. Beaches like Eristos to the south of Megalo Chorio have the odd basic taverna behind. In Livadia it will be worth going round the tavernas with an open mind as some are not exactly great.
I have not stayed in the hotel Irini but there are plenty of reasonably priced options in the vicinity.
There is no problem in going to both Ikaria and Tilos islands in a fortnight, starting by flights to Rodos, Kos or Samos.

Thanks Ouzo. Have you been to Ikaria - any views in comparison with Tilos? Not bothered if sand or pebbles, but a must are several tavernas, walks, secluded bays, etc with nooks & crannies to explore!