What sort of ranges are you looking at? If your ranges are, say, less than 100 yards for the smaller varmints and somewhat further for the larger ones then the .30-30 will work ok.
About lever action rifles and the rounds that they are commonly available in: Sadly, there are not as many offerings now as there once was. And some of the offerings that were once available are probably more appropriate for what you want to do. Nowadays, you have .22s, .30-30s, .35 Remingtons, .357 Mag (shoots .38 Spl as well), .44 Mag and some big bores.
Of these, the .22s are going to be a bit light on the larger critters. .35 Rem is more of a dedicated deer/black bear/hog round. .357 Mag/.38 is a possibility, but it doesn't perform all that well at longer ranges. .44 Mag is probably too much. And the big bores are intended for a completely different class of game than what you are intending to engage.
Which leaves .30-30. In its factory loadings, it is usually considered a deer round. But -- if you are handloader, you'll find that it will be better suited for the smaller varmint sized critters you expect to use it against. Normally, it is factory loaded with jacketed bullets in the 150gr-170gr range. But there are lighter jacketed bullets and cast bullets available to the handloader that should work well for what you want to do.