For those getting started- Ryan Kleckner writes a good book on long range shooting. Brian Litz is considered an expert on the subject, is a literal rocket scientist, and has his Applied Ballistics book which is excellent for those that want to get a little deeper.
I think the most helpful device in shooting long range today is the ballistics calculator. They are now downloadable as apps for your phone or tablet, and some are free. Once you fill in all the blanks, they will spit out the information in MOA, Mils, inches, and metric units if you would like. I personally like Strelok and they have a free version to try out. The calculator will need you to enter things like distance to target, wind and bullet velocity, ballistic coefficients, scope height over bore, and barrel twist rate. The further out you shoot, the more this information becomes critical. Garbage in equals garbage out so try to answer the questions the absolute best you can.
Next up is a good scope. I have shot to 1000 yards with a $200 scope but I much prefer being behind a really nice piece of glass. You will be using the scope to look at the target, measure hits and misses, and either dialing or holding to change the angle of the barrel for different distances/trajectories. I like something with at least a 16X zoom. There are many options for scopes these days, unlike when I got started where the choices were either a mildot or a duplex reticle. Now you can get reticles that look like a Christmas tree, MIL or MOA knobs and reticles, excellent zoom ratios, etc. Then comes MIL and MOA. Is Millradian better than Minutes of Angle? Each one is measuring angle, the difference is like inches are to millimeters, they each have a different scale. MIL has become more popular in recent years. The cost of the scope will translate into the quality of the product, how accurate it dials, clarity, and where it was made.
What rifle and caliber is needed to shoot past 100 yards? In my opinion, a rifle that will consistently shoot a 5 shot group under 1 inch at 100 yards, using match ammunition, under good conditions. Many bolt action rifles will do this out of the box these days. The caliber isn't super critical but some calibers make it much easier than others. I have shot to 1000 yards using a 223 and made pretty consistent hits. 308 was once very popular for precision shooting but has been overtaken by other calibers such as 6.5 Creedmore. The more you pay for a rifle doesn't always translate into more accuracy. A higher price usually comes with a nicer stock and trigger, and a smoother action. I have seen $500 rifles hit targets at 1000 yards but their level of comfort is lacking.
Ammunition is very critical to long range shooting. You can shoot pretty far with ball ammo but things will really start to fall apart around 600 yards. Match ammo is made with a higher quality bullet such as a Sierra Matchking, Berger VLD, Hornady ELD, etc. These bullets are built to tighter tolerances which help them fly more consistently.