Roman Girona Route

The best introduction to Girona is to go to Plaça Catalunya, 5 minutes walk from the apartment, and from there start the walk around the old wall of Girona (la Muralla). The views are magnificent and you will get your bearings. You look down on the old town and in the distance might see snow in the Pyrenees. 
Sometimes, about half way along, the door in one of the towers is closes and you have to retrace your steps. If so, find time if you can to go to the beginning of the other end of the wall, to be found near Sant Felix Church (Plaça de Sant Domenec) where you get another wonderful perspective on Girona old town and also some gardens to visit.

The question is, which Wall? There's the Roman one, the Carolingian or Medieval one, (that's the one described above) and then there's the Vauban-inspired fortifications on the left side of the Onyar river (following the Jaume I road). This last one is a guided tour from the Tourist office, and i shouldn't comment, never having done it, but it seems strange, as there is almost nothing left to see of it. It is extremely interesting if you can compare the present buildings with the old postcards of how it used to be, but as a walk, it is not pretty.
http://www.girona.cat/museuciutat/eng/recorreguts_muralla.php 
There are also lots of unkempt fortifications on the Montjüic mountain, which you can walk up or get the bus, but it's not a scenic walk at all. If you're interested though, i'll post more information about them.

The Roman Wall
Again which one? The first Roman walls of Gerunda, from 80-70BC, were surrounded by the 4 rivers - the Güell, Ter, Onyar and Galligants. They were built on the strategically positioned via Heràclia, later called via Augusta, by the Roman General Gneu Pompeu Magne. Then there were late 3rd century AD Roman walls which followed the same route. So it's the same thing as far as a walk goes. 
To tell the difference, look at the stonework. If you see large blocks of irregular or polygonal numulitic stone (Girona stone with little round fossils), it's from the original Roman wall. If you find large rectangular sandstone blocks (yellow), it's 3rd century.
Girona was founded by the Romans, despite being on the side of a hill, to defend the north south Via Heracles/Augusta route to Cadiz. That's all you're getting for now. Here's the route:

From the bottom end of the Rambla, c/ Argenteria, c/ Ballesteries, Pujada de Sant Feliu, pass Sant Feliu (Catalan)/Sant Felix (Spanish) church, don't go to the Cathedral, continue following the towers around the back of the cathedral, passing the Arab baths, going through the Jardins d'Alemanys, arriving at the Torre Gironella ruin - the highest point of the town. From here you continue along the top of the wall, going down to Torre del Telègraf. At this point go through the gateway underneath the tower and away from the university car park. You can see the Church of Sant Domènec and the university in front and the wall continues down on your right. Pass the church into the Plaça Sant Domènec (nice cafe under the tree shade, from where you can contemplate the Torre Rufina on the left).
Follow the steep steps down pujada Sant Domènec and turn right when you can to see the wall continuing up Escola Pia. You then retrace your steps a little to continue down to the Plaça del Correu Vell, where there used to be the 11th century Castell de Girona (no sign now). Only c/ de la Força in the Jewish quarter is left to do - the old cardo maximus of Gerunda.You are almost at your starting point here, once again. This encloses the 'Força Vella' and is the original Roman wall route, though with later Carolingian (9th-10th century) and medieval (14th century) on top. 

There are 3 visits you could add to the above to see most of Roman Girona.

1. In Sant Feliu/Felix church there are 8 white marble, Roman sarcophogi, mostly from early 4th century. They were found when the church was being built, in a necropolis along the via augusta/cardo maximus, outside the northern gateway to the city.

2. In Sant Pere Galligants church, there is the Archeological Museum which includes sepulcres from the mercadal necropolis ('el mercadal' was the town market situated at the bottom of the cathedral steps - just outside the Jewish 'call' or quarter.

3. In the Museum of History of Girona, c/ de la Força, there is the Mosaic of Can Pau Birol (3rd century).