East towards Italy
Catch the bus #6 to the stop Bon Voyage (costs EUR 1.00) or walk from the city towards the motorway signs (A8) along the river with no water, it takes about an hour. The motorway is above you, go to the onramp that says to Menton and Monaco. Plenty of room to pull over, and cars are going pretty slow. When going to Italy, wait for a car going at least to Menton (or use a sign), then ask to drop you out at the first petrol station (the only one between the two cities) shortly after the exit to Monaco, and look for drivers going to Italy there.
Take a train the bus No 10 to St Laurent du Var - Gare SNCF. From the station building north of the tracks walk along rue Léonard Anfossi. Soon you will arrive to a roundabout, follow on, pass under the motorway bridge, where you'll see a second roundabout, the exit on your left is the onramp for westbound A8. Use a sign saying Cannes and get off at the petrol station on the Cannes bypass.
Both west and East on the A8
There is a péage in nortwest Nice suitable for both directions. The problem though, is that it's difficult to access. Take the bus no.9 to Nice - La plaine and from there hitch or walk the remaining 2,5 km north along Route de Grenoble. Firstly, the motorway will be parallel to the road on your left, there you'll be able to see the southbound péage but unable to access it. A bit further is a roundabout with an on-ramp for A8 towards Monaco;Menton with a péage a 100 m further. To get to the southbound péage you'll have to keep north on Route de Grenoble (use the left shoulder), cross the A8 under the bridge and you will see a southbound on-ramp accessible only to cars coming from the north. Walk back along it till you get to the péage (although that's probably forbidden).
North towards Grenoble
That's a mountain road, where you can make it to Grenoble in one day or less (in summer). The first way to get there is the same as for the Péage on the A8, but from the roundabout just hitch towards Digne instead the A8. Other possibility is to take the "Chemin de Fer de Provence " train to Bellet, (at least 1 train an hour) the're is a good roundabout just next to the station.
If planning to go to the Mercantour National Park, you can take a public transport bus from Route de Grenoble that goes to Saint-Martin-de-Var and costs as the normal ticket. There is a roundabout just at the end of the village.
It is possible to pitch a tent on the beach west from the airport just across the river. A perfect hidden spot where you can find ducks, couples having picnics, kids drinking beer in the evening and some fishermen but they will all leave you alone. It can get crowded in peak season... To get there walk about 2 hours along the beach towards west until you pass the airport and cross the bridge. Go right immediately and walk along the river until it touches the sea. Or take the train to Saint Laurent du Var. There is a huge shopping mall on the beach. Behind it there are big rocks stacked in a wave defense wall. Cross it and enjoy.