Upvote:2
The Ethipoia Heritage Trust has a list of "Twenty One-day trips from Addis Ababa".
I'll leave you to read the full document, but the destinations by name are:
The document includes maps and details and descriptions of each trip.
In terms of transport however, while they say it's up to you, the first page does indicate it's worth considering taking a guard and driver. Possibly some of the locations are not that safe, or accessible by public transport. It does describe when you'll need a regular 'saloon' car or a 4WD.
Note that taxis and minibuses are considered public transport there anyway, and some rate them as the best public transport there, and they're cheap. So you really could hire a driver and get them to take you to one of them, or negotiate a day rate.
Of the list, Entoto sounds like the closest and most accessible, and certainly possible with other items on your own, with a taxi and / or public transport. An example of someone else doing that here.