@ "do I use the mean or mode": If you use validated questionaires they usually tell you in detail how to calculate the scale scores. If you have no instruction how to get get scores of the likert scales you can use a descriptive statistic that you consider meaningful for your data. This means if the likert scale scores shall show the central tendency of the items beloning to that scale you can use the mean or if you assume the data to be ordinal, you can use the median instead (which probably will be the case for likert scales). The mode can rather be used for nominal data as a statistic of central tendency. Of course there are other statistics you can use (see link from Ertxiem). In the end, you must decide what measure makes sense.
*@" I want to find the correlation between one likert scale and the rest of them": It also depends on the scale what correlation coefficient one has to use. For ordinal scale data you can use the spearman correlation.
Further, if you work with items you may want to read more about item analysis.