Get Gene Expression Matrix from GEOquery
The answer to this really depends on the type of data you're retrieving from GEO. Microarray data sets should have a normalised matrix of expression values uploaded as part of the entry. getGEO() from the GEOquery package will return a list of ExpressionSets - in many cases this list will have a length of 1. An ExpressionSet has an exprs() accessor for retrieving the expression matrix. An example (GSE76510, an Illumina HT12 v4 BeadArray dataset):

&gt; library(GEOquery)
&gt; gse76510 &lt;- getGEO('GSE76510')
&gt; names(gse76510)
[1] "GSE76510_series_matrix.txt.gz"
&gt; class(gse76510$GSE76510_series_matrix.txt.gz)
[1] "ExpressionSet"
attr(,"package")
[1] "Biobase"
&gt; exprs(gse76510$GSE76510_series_matrix.txt.gz) %&gt;% head
             GSM2027280 GSM2027281 GSM2027282 GSM2027283 GSM2027284 GSM2027285 GSM2027286
ILMN_1343291  14.936287  14.983927  15.008199  14.830549  15.008199  14.789385  14.891532
ILMN_1343295  13.404168  13.329903  13.289727  13.222221  13.386299  13.240755  13.125056
ILMN_1651199   6.779805   7.189868   7.046952   6.962957   6.697924   6.850950   6.792647
ILMN_1651209   6.983803   6.769702   7.016746   6.954751   6.890472   6.982947   6.915671
ILMN_1651210   6.837042   6.999098   6.895225   6.944250   6.955914   6.917959   6.837368
ILMN_1651221   7.034363   7.025626   6.606457   6.977622   7.051120   6.971884   7.066445
             GSM2027287 GSM2027288 GSM2027289 GSM2027290 GSM2027291
ILMN_1343291  14.891532  14.983927  14.848452  14.913474  15.008199
ILMN_1343295  13.240755  13.010757  13.525999  13.009661  12.998406
ILMN_1651199   6.786784   6.820451   6.851160   7.013586   6.864316
ILMN_1651209   6.983701   6.995199   7.270760   6.901377   6.950876
ILMN_1651210   6.804955   7.033074   6.894117   6.770691   7.000786
ILMN_1651221   7.003035   6.995199   7.050040   6.923861   6.929933


You will be able to filter out a list of required genes using the probe annotation (available in the appropriate annotation.db - in this case, for example, illuminaHumanv4.db).

If the dataset you're interested in is RNA-Seq, you're probably out of luck. There's currently no requirement on data submitters to provide expression estimates with an RNA-Seq dataset, so the ExpressionSet that GEOquery returns will most likely have an empty exprs slot.

Again, an example (GSE117133): 

&gt; gse117133 &lt;- getGEO('GSE117133')
&gt; exprs(gse117133$GSE117133_series_matrix.txt.gz)
 GSM3271892 GSM3271893 GSM3271894 GSM3271895 GSM3271896 GSM3271897