When going to Lugo, Spain, where else is ideal to visit, from there?

Where are you getting there from? In many cases, bus (coach) is the best option. Lugo has a wonderful Roman wall around the city, and it can be visited in one day. Like most of Spain, it´s not flat, but quite hilly. Guitiriz, Rinlo, A Coruña, Ourense...are at a normal driving distance for visits in one day. The area is totally safe and crime is extremely low, I would not worry the less about it.

Thank you for your information. Landing in La Coruna and driving to Lugo upon arrival.

Santiago de Compostela is definitely worth a trip. Similar distance from Lugo as A Coruna.

Are you arriving from North America after a night of no sleep? If so, instead of renting a car at the airport, take a bus from the airport to Lugo. You’ll need to transfer in A Coruña, taking approximately two to three hours. Rent a car after you recovered from jetlag that can take up to one or two nights.
Good day trips from Lugo are: https://en.wikivoyage.org/wiki/Main_Page.

• Santiago de Compostela (medieval center).
  
• A Coruña (historic center; direct bus [1h 30m]. There is also a direct train that takes longer).
• Ourense (geo-thermal springs; direct train [2h]).
• Ponferrada (direct bus [1h 30m]).

Above, Mikel mentioned Guitiriz that can be reached by direct bus (45m).

You may want to spend the day of your arrival or next day in A Coruña, then drive to Santiago de Compostela and then Lugo. Ourense is worth it, the termal baths by the river are a must. Lugo can be seen in one day, plenty of nature around it. You may want to drive to the coast up north, to Ribadeo, Rinlo and Playa de las Catedrales (depending on the time of the year, you need to book your entry here in advance). A Basque local, we spent last year 10 days in the area, stopping in Ribadeo, Rinlo, Foz and then on to Cedeira, our base to explore the beautiful, much ignored, north of Galicia, and then back by Puebla de Sanabria (one night), Lugo (one night) and the bath resort of Caldas de Luna on the way to León.

Mikel pretty much summed it up that a full day there is sufficient, probably more than is needed. Other than walking along/on the Roman wall that doesn’t take much time there are no other wow sites in the town. We walked around the town on the wall then again at street level. The town was serene in the evening and relaxing as there are few people wandering around. We stayed at Hotel Monumento Pazo de Orbán e Sangro. It’s just inside the wall and I believe it had its own parking area.

Cathedral beach that Mikel mentioned is best seen at low tide when one can actually walk beneath the rocks. For a nice view of the surrounding area, and in the same general area, is Cabo Ortegal. When we visited it years ago there were wild horses roaming around.

If you like old chapels, about 30 minutes from Lugo is Igrexa de San Salvador de Vilar de Donas. It dates from the 1400s. It is about a mile off the N-547. Puebla de Sanabria has a nice small castle that takes about an hour to see. It’s small main plaza has some architecturally pretty buildings. We stayed at the Paradors in both Ribadeo and Puebla de Sanabria.

Jaime, the wild horses are still there!! and Cabo Ortega's offers fantastic views, as is cape Estaca de Bares, the one dividing the Atlantic ocean and the Cantabrian sea.

Curious why you chose Lugo before investigating where to go near there (??)

You don't have to worry about crime anywhere in the region.