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To help manage our products through from initial quote, various production stages to invoicing and payment I am looking for a software platform that would allow different members of the team to access the project at relevant stages and take ownership and pass on to the next stage/person.
A colleague of mine is currently trying to shoehorn the various production processes into our sales CRM software and its making life very complicated for those involved in sales not production and those in production not involved in sales.
For simplicity I initially suggested a bloody great whiteboard and some coloured markers as they all sit in the same work space, however other more senior folk want to have software based solutions. (I've pointed out that this relies on staff access to said software via a form of hardware and the associated potential costs)
Anyhow, if you work in an SME with a sales and production based workforce and are currently using a system to help the days pass by smoothly, do let me know!
If a whiteboard would work for you, then using Trello would probably work too.
have a look at asana
I think it does everything you want
https://asana.com/?utm_source=unknown&utm_campaign=app.asana.com#close
+1 Trello and Asana if you must have a software solution
However, if you all sit together, the whiteboard solution is the best. What you're after is basically a Kanban board, which is what Trello and Asana provide in an (over-engineered for most applications, see below) electronic format.
 more senior folk want to have software based solutions
I see this a lot; people, especially and often senior people, when faced with an organisational issue around coordination and/or communication issue think "let's buy a software tool to fix it!". It's rarely the best option.
For a solution that will work, and that people will engage with (which is essential for it to work) ,remember the KISS principle:
Keep
It
Simple
Stupid