Once you do it, it’s no big deal. If you’re really self conscious, start small: have a coffee/bagel somewhere and chill – lots of people are eating solo at breakfast/lunch spots.
Sushi and tapas bars are great for solo dining. The environment is meant for lingering, your servers are more attentive, and more bang for the buck. Everyone pays attention to their own table anyway, so no one will notice your people watching or daydreaming.

