How to use Interpolation to fill in missing data

You have to provide the values of independent variable. Assuming that the points correspond to equidistant values of an independent variable, you can do this, for example:

int = 
 Interpolation[
   Select[Transpose[{Range[Length[data]], data}], NumericQ[Last[#]] &],
   InterpolationOrder -> 1
 ]

Of course, you may wish to scale the independent variable here in some way.

In versions 9+, you can also use the option MissingDataMethod after processing your data into a TemporalData object:

ClearAll[iF]
iF[d_, o_:1] := TemporalData[Replace[d, Except[_?NumericQ]:>Missing[], 1], Automatic,
   MissingDataMethod -> {"Interpolation", InterpolationOrder -> o}]["States"]

Examples:

data = {79., 79.1, 78.3, 78.6, 79., 77.2, 79.3, 79.3, "", "", 79.3, 76.6, 
79.3, 79.8, 78.4, 79., 78.4, 78.4, 79.4, 79.5, 79.4, 79.3, 79.2, 
79.2, 79.3, 79.6, "", 79.2, 79.3, 79., 79.1, 79.2}

Grid[Prepend[Transpose@{data, iF[data, 0][[1]], iF[data][[1]], iF[data, 2][[1]]},
 {HoldForm[data], HoldForm[intF[data, 0]], HoldForm[iF[data, 1]], HoldForm[iF[data, 2]]}],
 ItemStyle -> {Automatic, Automatic, {{{9, 12}, {1, 4}} -> Red, 
  {{27, 29}, {1, 4}} -> Orange}}, Dividers -> All]

style = {Joined -> True, Mesh -> Full, LabelStyle -> {"Panel", 14},
   ImageSize -> 300, PlotStyle -> Directive[{PointSize[.03], Thick}]};

Row[Table[ListLinePlot[MapAt[Style[#, Red, PointSize[Large]] &, dt[[1]], 
    Position[data, ""]], style, PlotLabel -> dt[[2]]], 
   {dt, {{data, "data"}, {iF[data, 0][[1]], "iF[data,0]"},
    {iF[data, 1][[1]], "iF[data,1]"}}}], Spacer[5]]