[color=black][size=5][center]Mental Sam[/center][/size][/color]
Center must be on the outside or the other codes will not work. Black text disappears in some templates, btw.
[center][b]and[/b][center]
The codes are in the correct order, but the 2nd tag is missing a /.
[color=indianred][size=5][center]RachelDrummond[/center][/size][/color]
[i][move]and many more!![/move][/i]
Move, like center, must wrap around all other codes.
When you center 3 lines in a row, repeating the tags on each line isn't necessary. The corrected coding would look like this:
A very [b]Happy & Healthy[/b] Birthday to
[center][color=black][size=5]Mental Sam[/size][/color]
[b]and[/b]
[color=indianred][size=5]RachelDrummond[/size][/color][/center]
[move][i]and many more!![/i][/move]