Jessie and James are the key members of the infamous Team Rocket, and more often than not, they will be involved with whatever Team Rocket is up to. Believe it or not, you will battle Jessie and James more during the game than you will both Archer, the Admin of Team Rocket, and Giovanni, the Boss of Team Rocket. Actually, you will battle Jessie and James as many times as both Archer and Giovanni combined!
You will first run into Jessie and James in Viridian City at the start of your journey; however, you won't be formally introduced to them until they stop you in Mt. Moon. They will demand you hand over the fossil you have obtained directly before the exit of Mt. Moon, and they will challenge you to a battle when you refuse. They will battle using Ekans and Koffing - both at Level 12 for this encounter - the only time they will battle with these two Pokemon before evolving them into Arbok and Weezing, respectively.
The next time you will see Jessie and James is in Lavender Town after visiting the Pokemon Tower, but they will quickly head to the Game Corner in Celadon City. You will again see them in the Game Corner, but they will be too busy playing slots to notice you. Honestly, you won't battle Jessie and James until near the end of the Rocket Hideout, which is found just below the Celadon City Game Corner. They will battle you with Arbok and Weezing this time around, both at Level 32. From there, you will battle Jessie and James at the top of Pokemon Tower when trying to save Mr. Fuji. They will continue to use Arbok and Weezing in battle, this time each at Level 34.
After defeating them in Pokemon Tower, odds are you will next find Jessie and James in Saffron City when entering the Silph Co Building. However, you won't battle them until you have reached the eleven floor of the structure. Their Arbok and Weezing will each be at Level 36 for this battle. Next, you will run into Jessie and James just south of Fuchsia City on Route 19, when they will give you the Gold Teeth they found when searching for treasures on the beach. They will not battle you during this encounter.
You will not see Jessie and James again until on Route 17 after you have defeated the Elite Four. You will overhear them talking about how Giovanni is in hiding since Team Rocket disbanded, and they will challenge you to one last battle when they notice you listening in. Both their Arbok and Weezing will be at Level 52 for this final battle. After defeating them for this last time, they will again offer you to join Team Rocket. After you refuse, they will give you the Blast Off Outfit just in case you ever change your mind.