From Jesenice train station you can go on the biking track, but as it goes quite up and down, you can perhaps better take the main road and go on the bike track on the bridge before Mojstrana. Then you follow it (as on the map) through Kranjska gora and further on to Tarvisio.
By Alpe-Adria (signposted 'FVG1') you ride all the way till Pontebba. From just before the town (when you are under the highway), you can consider taking the ordinary road into the town - it's much faster!
From Pontebba on, you simply follow the 'Pontebbana' ex railway. Only in few places be careful not to miss specific passages. They are well marked (plates), just be attentive! In some tunnels having lights is very useful - there are some quite dark passages! Otherwise the scenery in this valley is great. The last two tunnels (2017) past Resiutta are long, but nicely lit, then the 'Pontebbana' stretch ends.
One option to continue is to ride across the river towards Moggio Udinese and then just before the town turn left and ride by the roads on the right side of Fella river. Some stretches are not asphalted there! You return back on the left side of the river by the ex railway bridge at Carnia village. This detour is much longer and time consuming, so it's more recommendable to ride from Moggio till Carnia and further to Venzone by the main road (quite busy).
From Venzone, you can continue towards Gemona (as the 'medieval' Venzone, also a very nice town!) and then more directly towards Udine - depending where you want to sleep. The proposal here is however to continue along the beautiful Tagliamento/Tilment river till San Daniele. So, you cross the river in Venzone and ride by the right side through the villages. Before Cornino, you ride past a small lake (natural reserve) and then on the village turn left and cross the river by the big bridge. Then either continue directly to San Daniele, or ride gently past the small Lago di Ragogna.
Day 2 is difficult to describe in detail. The route depends on what you want to visit. In the Upper Rive dArcano there's an interesting castle, Fagagna is sitting on a hill, but it's worth riding up to the church and castle. Then you can descend directly towards Udine and explore the town, or you keep riding north of it. The road through Moruzzo and Pagnacco goes more up and down, but the landscape is nice.
On the other side of Udine, you ride towards the south-east. In Buttrio ride by the country road north of the main road, from Cormons you will need to follow more or less the main road. No biking tracks there! Finally, you need to ride through the whole town of Gorizia (that's still Italy), and only in its northern part cross the border with Slovenia. Immediately as you are over it, you will get on the railway station.