What is the best online Project Management software for a startup?

I personally use Opengoo, please check out the list from here, http://www.cyberciti.biz/tips/open-source-project-management-software.html (They are all opensource and should be free) there's no use me talking about it more or less , I'd say give it a shot and see what suits your purpose the best, Cheers!

Use MS Project as a starter and then move to Primavera if you are in the Construction field.

There are hundreds of them out there, so it really depends on what you are looking for? What kind of projects, project scope ect, you should then specify your claims, what is it that you need to manage time, monitor project or make the job easier.
Be aware that there is no defined definition for project management software because it alters.

Online project management software cleaves into two halves: 1.
The management of Project-Calendars, Milestones, Tasks, and To-DOs and 2.
The online communication between team members, freelancers, and vendor-whiteboards, forums, chat and messaging systems.
Some good online Project Management tools are, a.
Ace Project b.
Zoho Project c.
Base Camp d.
Huddle There are two presiding philosophies to online project management: One says, "Less is More" and the other says, "More is More." On the more is better side of the spectrum, Ace Project and Zoho Project are the finest options I encountered.
Both have online storage, plenty of (paid) options to grow your business, and can be customized to the core.
Whereas Ace is high when it comes to free projects, storage, and tracking functionality, Zoho serves up Google Docs integration and an alphabet of features and ancillary products.
Hope this helps..
Regards.
ZA

Check JIRA

usually start up projects are relatively small , and it doesn't need any project management software , i prefer to use the time on developing the product rather than spending time on data entry and such things , usually you are alone in developing the project and you dont need to report and to c who is who and that staff , i used to use project management software while i was working as an employee , after i had my own startup i didnt need it any more

Basically,There are lots of project management software are available but you cannot reach them with a single path so if you want project management software with quality data then you should visit Proofhub that helps you and your organizations. It has gantt chart, milestones, to-do's, events, calendar, chat, discussions, file sharing, time and progress tracking and much more. It is integrated with google docs. and supports all cloud functionality. UI is user friendly and much better than any other PM software. Have a look http://www.proofhub.com/

As a start up I'd recommend a free project management software with premium features - try ProjectToolBelt

ProjectToolBelt is fully free for unlimited users and projects - it has project and task management (with sub tasks and recurring tasks) as well as To-Dos and milestones. Estimated and Actual Costing and budgeting features make it easy to do project costing. It has one of the best search features to find anything quickly -  with the Time Filter and Quick Filter. Communication between team members and vendors or customers can be done using the Notes, Discussions and all auditing and activity can be viewed in the activity streams. Counters and graphs give a visual view of project progress.