I made this Connect 4 game twice. Once in 2010 and again in 2021. The first time I used MDF, which allowed me to cut very smooth holes. Then I sliced up a dowel for the playing pieces and painted everything. For the second one, I used solid walnut and drilled out the holes using a backer board which reduced splintering a lot. 

Either way, download this PDF template and use spray adhesive to attach it to the work pieces. Then simply drill out all the holes using a drill press and the correct Forstner bits. You could use a hand held drill, but it will be a bit more difficult. 

Please watch the videos for more instruction!