Simply put, the .308 is an order of magnitude "more powerful" than the 30/30.
Standard loads for the 30/30, using a 170 grain bullet for example, will produce a muzzle velocity of about 2100 fps....with impact energy to match.
Standard loads for the .308, using a 168 grain bullet for example, will produce around 2800 fps.....with FAR greater impact energy.
Recoil will also be much greater with the .308, depending on the load.
Both 30 cal. rounds.....but two VERY different cartridges.
If you are considering getting into long range shooting, there are lots of good choices, depending on your needs. 1000 yard shooting, with any accuracy is, of course, the top of the heap in terms of equipment required. You'd be far better off if you set your goals to a bit more modest level, to start. For instance, 300-400 yards will still give a good challenge, but be easier to attain (and far less costly).
As for cartridges, you could go with anything from the .223, on up to large, much more exotic rounds, such as the .338 Lapua.
Starting out, with distances out to 300 to 400 yards maximum, I'd recommend the .223. Very accurate, flat shooting, not as expensive as many others, very common......and little recoil. There are lots of other cartridges that would do, however.