Your solution is slow because the series expansion is done anew for every numerical value of x, I suppose. How about this:

Clear[x, n, f];
nmax = 10;
Do[f[x_, n] = Normal[Series[Exp[x], {x, 0, n}]],
 {n, 0, nmax}]

where nmax is largest n that you are going to use.

Another way:

g[x_, n_] = Sum[SeriesCoefficient[Exp[x],
    {x, 0, i}] x^i,
  {i, 0, n}]

The same problem happens with the derivative:

f[x_] := D[x^2, x];
f[1]

Regarding the proposal of Bill: indeed the Plot works, and a command

F[x]

produces an expansion formula, but there is a problem calculating a value for a particular x, for example:

Evaluate[F[2.0]]

I have tried:

F[x_]:=Module[{y,c},c=Normal[Series[Exp[y],{y,0,2}]];y=x;c];

and this seems to work correctly in all three cases, that is one can show a formula, compute a value, and create a plot without using Evaluate[]. Is this an acceptable approach? Can anybody explain to me what is going on here? I cannot understand the mechanism, and the information contained in manuals is very limited. Leslaw

 F[x_]:=Normal[Series[Exp[x],{x,0,2}]];
 Plot[Evaluate[F[y]],{y,0,1}]

<<<and your plot appears>>>