If you're at all able to start with something approaching the business-facing parts of Scrum, I think that's a great place to start. In Scrum, there are ideally two completely separate queues: The product backlog belongs to the business folks, and is organized in a way that works for them, and the dev team's list of tasks belongs to dev, and probably shouldn't look similar at all. And the only way for things to move from one queue to the other is during a regular meeting where the dev team says, "OK, it looks like we've got room to take on X additional work, what can you give us?"
And then, and I think this is secretly one of the most important things, the dev team physically creates new tickets for the new work they're taking on. You should never be able to just drag-and-drop tickets from one queue to the other. Ideally, it shouldn't even be physically possible to do so. Where I am right now, one is in $JIRA_COMPETITOR, and the other is an Excel spreadsheet. (If we all worked out of the same office, I'd probably go for an Excel spreadsheet and yellow sticky notes on a wall. Yellow sticky notes on a wall are awesome. They're a great deterrent to keep the PHBs from generating reports and dashboards for bothering people with KPIs.)
This little ceremony, and the associated firewall between two related but separate business functions, is, I think the heart and soul of what Scrum should be. It's the thing that sets up all the little firewalls and incentives that encourage business and dev to maintain a relationship that's more functional than dysfunctional. Everything else - sprints, story points, standup meetings, all that jazz - is just window dressing that some teams may or may not find makes that ceremony go more smoothly.
And if you can get a formal handoff like that in place, and get people to respect it, my (admittedly hopelessly idealistic) belief is that the rest of it gets easier to figure out. What's the correct scope and size of individual requests that the product people make of the dev team? Whatever size and scope best enables people to come out of the meeting feeling good about how it went.
But if everything's being jammed into one gimongous ticketing system, like all the ticketing system vendors seem to want you to do, you're doomed.