Beginner code for a Rock, Paper, Scissors game in Python
The code repetition could be cleaned up quickly by replacing your seven if blocks with something like:

print("Your opponent played: " + str(opponentMove) + ".")

if player1Move == opponentMove:
    print ("Tie!" + " Please go again.")
elif (player1Move == "Rock") and (opponentMove == "Paper"): # rock loses to paper
    opponentScore = opponentScore + 1
elif (player1Move == "Paper") and (opponentMove == "Rock"): # paper beats rock
    player1Score = player1Score + 1   
elif (player1Move == "Paper") and (opponentMove == "Scissors"): # paper loses to scissors
    opponentScore = opponentScore + 1
elif (player1Move == "Scissors") and (opponentMove == "Paper"): # scissors beats paper
    player1Score = player1Score + 1
elif (player1Move == "Scissors") and (opponentMove == "Rock"): # scissors loses to rock
    opponentScore = opponentScore + 1
elif (player1Move == "Rock") and (opponentMove == "Scissors"): # rock beats scissors
    player1Score = player1Score + 1
else:
    print ("Invalid choice")

print("Your score: " + str(int(player1Score)))
print("Your opponent's score: " + str(int(opponentScore)))


This eliminates all of the continue statements.  Keeping with the same approach you have started, perhaps splitting into cases where player1 wins vs opponent wins is the next improvement:

if player1Move == opponentMove:
    print ("Tie!" + " Please go again.")
elif ((player1Move == "Rock") and (opponentMove == "Paper")) or
     ((player1Move == "Paper") and (opponentMove == "Scissors")) or
     ((player1Move == "Scissors") and (opponentMove == "Paper")):
    player1Score = player1Score + 1
elif ((player1Move == "Paper") and (opponentMove == "Scissors")) or
     ((player1Move == "Scissors") and (opponentMove == "Rock")) or
     ((player1Move == "Rock") and (opponentMove == "Paper")):
    opponentScore = opponentScore + 1
else:
    print ("Invalid choice")