If you want to go to Rome to visit amongst others the aqueducts, please remember nowadays Rome is very busy, big and full of cultural sites. That's why you will find tourists everywhere in every season. Furthermore, Roman traffic can be hazardous. Traffic rules do not apply to all those scooters / motor cycles who flock the city, especially in the morning and afternoon.

Most Romans do not know where the aqueducts are (our experience). Maps are also not very helpful, but the famous Roman fountains can be found rather easily on them. These can help you pinpointing the aqueducts.
Trust this and my under mentioned suggestions.

Suggestion 1: Parc of Aqueducts (Parco degli Acquedotti)
How to get there
Metro line A, via Termini Station (last station Anagnina). Leave the metro at stop Giudio Agricola and go upstairs, follow the Via Giudio Agricola till you see the church (Chiesa), to the right and behind the church you will find the entrance to the parc (see the sign left).

What to see
The remains of 6 aqueducts, i.e. Marcia, Neo Anio, Tepula, Gulia, Felice en Claudia. The parc is not that good maintained but the monuments are great. In May many flowers were visible round the aqueducts, good for the pictures. Between two aqueducts, Aqua Felice and Aqua Claudia, you can find several allotments (gardens). To visit Aqua Claudia up close you can also follow this route: leave the church and cross the first aqueduct (Felice), turn right. Follow this aqueduct till you reach the end of the allotments (gardens), turn left. This way you will reach a concrete plateau. From here a path can be followed through the pasture, to Aqua Claudia.  See the photographs.

Tip 2: Trevi fountain / Aqua Vergine
How to get there
On foot, it is near the Via del Corso. Following the Via delle Muratte you will reach the square with the fountain. Somewhat before the square you can see the aqueduct (Aqua Vergine) behind a fence, in Via de Vergine.

What to see
Of course this monumental fountain. The name derives from the word Trivium, it is a place where three roads meet. Please remember that in 19 before Christ Agrippa built this fountain to bring "aqua virgo" (clean, drinkable water) to Rome. See the photos.
Why "aqua virgo/virgine"? See legends.
This aqueduct brings water to several fountains in the historical centre of Rome, from the Piazza Navona up to the Piazza di Spagna (famous and reachable by metro, line A, stop Spagna).

Suggestion 3: Fonte Acqua Paola / Monumento a Garibaldi and several parcs
How to get there
On foot from Trastevere; via S. Maria in Trastevere, Via Garibaldi and the Piazza S. Petro in Montorio.

What to see
First of all a great view over Rome opposite of the fountain, the end of the aqua Paola. There is not much to be seen of the aqueduct. A sign tells us that we have found it, but the fountain is again very monumental. See the photos.
From here up the hill you will find the Garibaldi-monument and the Foro della Vittoria (lighthouse for victory). In this parc (Parco Gianicolense) Roman people enjoy the view and the weather in the weekend. Recreation parcs: amongst others Villa Doria Pamphili and Villa Abamelek.