Please help: I know that compounds are two or more different elements bonded. While on the other hand, elements aren't bonded. (Not so sure however.)

Answers :
 a₀ = 3
 a₁ = 1
 a₂ = 1
 a₃ = 3

Explanation :
 
To balance the reaction equation, both left and right hand sides should have same number of atoms in each element.

Given reaction is
a₀HClO₄ + a₁Pb(NO₃)₃ → a₂Pb(ClO₄)₃ + a₃HNO₃

Here, 
 Left Hand Side has                                         Right Hand Side has
   H = 1 atom                                                         H = 1 atom
   Cl = 1 atom                                                        Cl = 3 atoms
   O = 13 atoms                                                     O = 15 atom
   N = 3 atoms                                                       N = 1 atom
   Pb = 1 atom                                                      Pb = 1 atom

Pb is already balanced. Then others should be balanced by adding coefficients.

First step : balance the N atoms in both sides.  
There are 3 N atoms in left hand side and 1 N atom in right hand side. Hence, to balance N atoms "3" should be added before HNO₃. 
 
Second step : balance the Cl atoms.
There are 3 Cl atoms in right hand side while 1 atom in left hand side. So, "3" should be added before HClO₄ to balance Cl atoms in both sides.

After doing those, number of O atoms and H atoms in both sides is automatically balanced.