A car is pretty much a necessity if you're going to be taking day trips from Oban. There are limited long haul buses out of Oban - only three per day heading north toward Fort William, and limited service south toward Kilmartin, Kennacraig, and Campbeltown. If you wanted to visit someplace like Glencoe, you'd need to take two different buses. The train will take you back to Tyndrum, where you can catch a connecting train to Fort William. The Tyndrum to Fort William train runs across Rannoch Moor, and is one of the many scenic train journeys in Scotland
Without a car, you could still catch the ferry to Craignure, take a bus to Fionnphort, and take the passenger ferry to Iona. You could also take a bus up to Tobermory, which is charming, but again, there's not a whole lot to do. However, as Allan pointed out, if you're looking for a "quiet day," Tobermory would be a better choice than Oban. If you have a car, you could visit the hamlet of Calgary, which has one of the prettiest beaches on the islands.
If you decide to spend a couple of days in Oban, you could visit the islands of Kerrera and Lismore, both of which have excellent walking/hiking. Kerrera is the closest to Oban, and can be reached by a passenger ferry. Lismore is a bit farther away, but would make for a good day trip. Last I heard, they still had bicycle rental near the ferry terminal. There's virtually no traffic on Lismore, so bike riding is safe.
If you're going to wait until you get to Oban to hire a car, be aware that there are only a couple of small car hire companies in town, and because they are small independent companies, you'd need to bring the car back to Oban. Also, because they have the Oban market cornered, they are not going to be cheap. You need to have no worries about being able to take a hired car on to the ferries. You can't get around the Highlands and Islands without taking ferries. Scottish car hire companies understand that.
If you have a car, I'd recommend one or two nights in Oban; the ferry to Craignure; and one or two nights in Fionnphort, so that you can have a full day on Iona and a boat trip to Staffa. From Fionnphort, you could drive up to Tobermory, spend an overnight, then take the Tobermory to Kilchoan ferry. From Kilchoan, you can follow Jennifer's advice, and visit the western end of the Ardnamurchan Peninsula. After you've visited Ardnamurchan Point and Sanna Bay, you'd have the choice of heading back along the peninsula, then north from Salen to meet the A830, which would give you the choice of heading west to Mallaig and Skye, or east to Fort William via Glenfinnan. Your other choice from Salen would be to head directly east along Loch Sunart, and take the Corran Narrows ferry back to the A82 south of Fort William.
Mull does not have as complex a road system as Skye does. Almost all of the roads are coastal, and the minor roads are spectacular.
Whatever you decide enjoy your time in Scotland!
Best wishes, and stay safe.