Give a try to ProofHub, a great alternate to Asana. It’s much more feature rich and user friendly. With ProofHub, your projects, teams, clients and all the communication stays in one place without the need of investing in too many different tools to run your business. This makes it easy to create plans, collaborate with teams and clients, keep things organized and deliver projects on time.
Here is the link: https://www.proofhub.com
Many companies go with Atlassian's JIRA because of wide adoption, large degree of customization, built in issue query language (JQL), and integration into other good Atlassian products such as their wiki Confluence, or their VCS BitBucket.
Commercial companies ok with not self-hosting their issue tracking have other options. One of the better ones is Trello, made by FogCreek software, the same folks that brought you Stack Overflow. They also have a self-hosted solution called FogBugz that is very high quality from what I've heard, but doesn't have a huge amount of adoption in the companies I've worked at.
One thing to consider is whether the product matches your workflow. Are you more Kanban, or Scrum, or something else? Trello is great for a Kanban style issue management, but doesn't have the issue track specific advanced features that JIRA does, e.g. JQL.