Legendre Polynomial 
w(x)=1, (a,b)=(-1,1)

ϕ[n_,x_]=LegendreP[n,x];
w[x_]=1;
a=-1;
b=1;

num=11;
plota=a;
plotb=b;

Polynomials & Standardization

TableForm[Table[{n,ϕ[n,x],ϕ[n,1]},{n,0,num}]]

ϕ[n,1]

Normalization

TableForm[Table[(∫_a^b)ϕ[n,x]ϕ[m,x]w[x]\[DifferentialD]x ,{n,0,num},{m,0,num}],TableSpacing->2]

Zeros

TableForm[Table[{n,SetPrecision[NSolve[ϕ[n,x]==0,x],10]},{n,1,6}],TableSpacing->1]

Plots

Do[plot[n]=Plot[{ϕ[n,x]},{x,plota,plotb},PlotRange->{{-1,1},{-1,1}},PlotStyle->{Red,Thick}],{n,0,num}]
Table[{plot[n],plot[n+1],plot[n+2]},{n,0,num,3}];
Show[GraphicsGrid[%],ImageSize->{650,Automatic}]

Laguerre Polynomial 
w(x)=(e^(-x)), (a,b)=(0,∞)

ϕ[n_,x_]=LaguerreL[n,x];
w[x_]=Exp[-x];
a=0;
b=Infinity;

num=8;
plota=a;
plotb=10;

Polynomials & Standardization

TableForm[Table[{n,ϕ[n,x],ϕ[n,1]},{n,0,num}],TableSpacing->1]

Normalization

TableForm[Table[(∫_a^b)ϕ[n,x]ϕ[m,x]w[x]\[DifferentialD]x ,{n,0,num},{m,0,num}],TableSpacing->2]

Zeros

TableForm[Table[{n,SetPrecision[NSolve[ϕ[n,x]==0,x],10]},{n,1,6}],TableSpacing->1]

Plots

Do[plot[n]=Plot[{ϕ[n,x]},{x,plota,plotb},PlotRange->{{0,5},{-2,2}},PlotStyle->{Red,Thick}],{n,0,num}]
Table[{plot[n],plot[n+1],plot[n+2]},{n,0,num,3}];
Show[GraphicsGrid[%],ImageSize->{650,Automatic}]

Hermite Polynomial 
w(x)=(e^(-(x^2))), (a,b)=(-∞,∞)

ϕ[n_,x_]=HermiteH[n,x];
w[x_]=Exp[-x^2];
a=-Infinity;
b=Infinity;

num=8;
plota=-2;
plotb=2;

Polynomials & Standardization

TableForm[Table[{n,ϕ[n,x],ϕ[n,1]},{n,0,num}],TableSpacing->1]

Normalization

TableForm[Table[(∫_a^b)ϕ[n,x]ϕ[m,x]w[x]\[DifferentialD]x ,{n,0,num},{m,0,num}],TableSpacing->1]

Zeros

TableForm[Table[{n,SetPrecision[NSolve[ϕ[n,x]==0,x],10]},{n,1,6}],TableSpacing->1]

Plots

Do[plot[n]=Plot[{ϕ[n,x]},{x,plota,plotb},PlotRange->{{plota,plotb},{-100,100}},PlotStyle->{Red,Thick}],{n,0,num}]
Table[{plot[n],plot[n+1],plot[n+2]},{n,0,num,3}];
Show[GraphicsGrid[%],ImageSize->{650,Automatic}]

Chebyshev Polynomial of the First Kind 
w(x)=(1/(Sqrt[1-(x^2)])), (a,b)=(-1,1)

ϕ[n_,x_]=ChebyshevT[n,x];
w[x_]=(1-x^2)^(-1/2);
a=-1;
b=1;

num=8;
plota=a;
plotb=b;

Polynomials & Standardization

TableForm[Table[{n,ϕ[n,x],ϕ[n,1]},{n,0,num}],TableSpacing->1]

Normalization

TableForm[Table[(∫_a^b)ϕ[n,x]ϕ[m,x]w[x]\[DifferentialD]x ,{n,0,num},{m,0,num}],TableSpacing->1]

Zeros

TableForm[Table[{n,SetPrecision[NSolve[ϕ[n,x]==0,x],10]},{n,1,6}],TableSpacing->1]

Plots

Do[plot[n]=Plot[{ϕ[n,x]},{x,plota,plotb},PlotRange->{{plota,plotb},{-1,1}},PlotStyle->{Red,Thick}],{n,0,num}]
Table[{plot[n],plot[n+1],plot[n+2]},{n,0,num,3}];
Show[GraphicsGrid[%],ImageSize->{650,Automatic}]

Chebyshev Polynomial of the Second Kind 
w(x)=(Sqrt[1-(x^2)]), (a,b)=(-1,1)

ϕ[n_,x_]=ChebyshevU[n,x];
w[x_]=(1-x^2)^(1/2);
a=-1;
b=1;

num=8;
plota=a;
plotb=b;

Polynomials & Standardization

TableForm[Table[{n,ϕ[n,x],ϕ[n,1]},{n,0,num}],TableSpacing->1]

Normalization

TableForm[Table[(∫_a^b)ϕ[n,x]ϕ[m,x]w[x]\[DifferentialD]x ,{n,0,num},{m,0,num}],TableSpacing->1]

Zeros

TableForm[Table[{n,SetPrecision[NSolve[ϕ[n,x]==0,x],10]},{n,1,6}],TableSpacing->1]

Plots

Do[plot[n]=Plot[{ϕ[n,x]},{x,plota,plotb},PlotRange->{{plota,plotb},{-4,4}},PlotStyle->{Red,Thick}],{n,0,num}]
Table[{plot[n],plot[n+1],plot[n+2]},{n,0,num,3}];
Show[GraphicsGrid[%],ImageSize->{650,Automatic}]

References

http://mathworld.wolfram.com/OrthogonalPolynomials.html