I highly suggest relying on taxis and public transport
The metro is the fastest way to move around in Paris. However is can get pretty chaotic during rush hours (8am to 10am and 5pm to 7pm)
You can download the official Metro app (iPhone / Android) which will allow you to view itineraries to move around using the metro. You also can use Google maps which works as well to calculate itineraries using public transports.
If you plan on using the metro, the best way is to buy a rechargeable card (called "Passe Navigo Easy"). The card costs 2€ and can be bought at most metro stations. Once bought, you can recharge the card directly with your smartphone using the metro app mentioned above.
If you don't like the metro there is also a bus network but it tends to be slower.
Train / TGV
Trains are greats if you want to travel outside of Paris. The connections are pretty fast. You can go to Lyon in 2h, to Marseille by the mediterranean sea in 3h. You also can connect to international cities like London or Brussels using the train.