The first question I would ask is: What is available in your area? Check the local clubs and ranges and see what events are being shot. Then visit and see what guns are in use.
Standard metallic silhouette is shot at ranges of 200 to 500 metres. Your .270 would get you started there, but you would likely want something specialized before long. Last time I looked which was several years ago, the 7mm 08 was popular in rifles set up to just barely meet the rules.
Conventional and F-class long range start at 600 yards and run out to 1000. Again, your .270 would enter in an "any sight" or F-class long range match but I don't know how well it would hold up. I have shot in F-class (prone with artificial support allowed) with my old .30-06 and had a good time. Most of the guys had very nice (expensive) .308s, 6.5x284 or 6mm match rifles.
Across the course, 200, 300, 600 yards is best shot with a very good AR these days.
But try to check out some actual matches in person, I can't tell you everything you need to know on the Net, partly because there isn't time or room enough, partly because I am not a major league long range shooter.
Your .30-30 and .300 Savage are not well suited for any competitive target shooting that I know of.