If you want to own a centerfire rifle with a faster rate of fire, you need a rifle with semi-automatic capabilities. Unlike bolt action, lever action, or pump action rifles, semi-automatic rifles will fire a bullet with every pull of the trigger (assuming that the round has been chambered first, of course).
Semi-Automatic rifles first became a thing in the beginning of the 1900s, and have steadily and consistently evolved since then. Available for both military and civilian use (at least in the United States), semi-automatic rifles are also versatile because of the fact that they can be used for both hunting and tactical/defense use.
Vote below for the best semi-automatic rifles of all time:
"The greatest battle implement ever devised," said General George S. Patton in regards to the .30-06 caliber M1 Garand. This rifle was the standard issue infantry rifle of World War II for the United States military, and provided the American soldier with a definitive advantage over their German, Italian, and Japanese counterparts due to its semi-automatic firing capabilities and eight round magazines (Axis troops were almost universally armed with five round bolt action rifles). The M1 Garand also proved to be a highly popular rifle after the war, remaining in service until the late 1950s when it was replaced by a modified version called the M14/M1A. Even then, it remains a popular rifle as a collector's and historical piece. Unfortunately, no manufacturers make the M1 Garand today (because it would be expensive to produce) so for the time being you'll have to settle for a military surplus rifle.
The M14 (or the civilian M1A) is essentially a modified M1 Garand designed to be more modern. It was adopted by the United States Army as the M1 Garand's replacement in the 1950s, and has continued to serve in limited use as a designated marksman's weapon (DMR) to this very day. Basically, the M14 is an M1 Garand that has a slightly shorter action to accommodate the shorter .308 Winchester round, a detachable box magazine (which holds anywhere from 5-20 rounds depending on the magazine), and a shorter gas tube that doesn't run nearly as long as the barrel. Other than those differences, the M1 Garand and the Springfield M1A are essentially the same weapon. The M1A is a highly versatile weapon due to the fact that it fires the common .308 round, is powerful enough to drop big game animals, is semi-automatic and therefore good for tactical or defense use, and can shoot rounds out to very long distances.
The SKS was developed at the end of World War II and, contrary to what most people think, actually saw very limited service in the final stages of the Eastern Front. However, while the SKS may have been meant to replace the Mosin Nagant as the standard issue infantry rifle of the Soviet Union, it was ultimately overshadowed by the AK-47 rifle instead. The SKS is a semi-automatic rifle that fires the 7.62x39mm round (the same round as the AK-47 with ballistics similar to a .30-30) and is located via a stripper clip in the top of the receiver. The SKS was designed to be a shorter rifle firing an intermediate cartridge with a faster rate of fire than the Mosin Nagant, but the fact that the AK-47 had greater capacity and a detachable box magazine are what ultimately caused it to be adopted instead. Nonetheless, the SKS was adopted in limited use by the Soviet Union and was sold in mass quantities to Communist countries through the Cold War. Today, it's a common military surplus weapon in the United States.
The HK G3, often sold to American shooters today in the form of clones like the PTR-91, CETME, and Century Arms C308 rifles, was designed as a battle rifle in the 1950s and became known as one of the Big Three battle rifles along the FN FAL and the M14. It became the standard issue service rifle of the German Army until it was replaced by the HK G36 and HK 416 rifles, and was also issued as the main rifle to many other countries as well. The manual of arms of the HK G3 is very similar to that of the HK MP5 - indeed, the MP5 is essentially a cut down G3 designed to fire pistol caliber rounds like the 9mm. The G3 comes standard with a 20 round magazine and fires the .308 Winchester round (or the 7.62x51mm).
Known as the 'Right Arm of the Free World,' the FN FAL is one of the most widely issued military service rifles in history. Through the Cold War, whereas most Communist bloc continues were issued the AK-47 and SKS, most of the 'free' Western countries were being issued the FN FAL (hence the name). The FAL is also widely available today as a semi-automatic rifle to American shooters, although it is less popular than the Springfield M1A. The FAL comes standard with a twenty round magazine and fires the .308 Winchester round.
Of course, you probably expected to see the AR-15 on his list. The AR-15 is the most popular rifle in the United States and also long been at the center of controversy politically speaking. The AR-15 is basically a semi-automatic only version of the military's M4 and M16 carbines and rifles, which are selective fire. It fires the 5.56x45mm NATO round usually, but is also chambered for a number of other cartridges as well. ARs that fire the .308 Winchester or other intermediate rounds are referred to as the AR-10. AR-15s are known for being easy to shoot, ergonomic, and with limited recoil. They can be a great choice for tactical training, home defense use, or hunting.
Another rifle that you probably no doubt expected to see on this is the AK-47. Technically, the AK-47 can be either a selective fire or a semi-automatic only version. In the United States for civilians, most AKs sold are semi-automatic only for legal reasons. The AK-47 usually fires the 7.62x39mm round, the same round as the SKS, and comes standard with a thirty round magazine. The 7.62x39mm offers superior power to the 5.56 of the AR-15, and is therefore a superior round to use for hunting larger game. Contrary to what many people think, the AK-47 can indeed be used as an effective hunting weapon if necessary, and it can also easily be used for home defense or tactical use as well.
If you're looking for a 5.56 rifle but don't want an AR-15, your next best choice would arguably be the Ruger Mini-14. The Mini-14 is essentially an M1A that has been scaled down to fire the 5.56 round, and therefore has an essentially identical manual of arms. Ruger makes numerous versions of the Mini-14 today, including some that can fire either the 5.56 or the .223, others that can just fire the .223, others that fire the 7.62x39mm, some with a wood stock or synthetic stock or folding stock, some with a flash hider and others without it, and so on.
Common knowledge amongst the firearms community is that the standard issue infantry rifle of the Soviet Union during World War II was the Mosin Nagant in 7.62x54r. It was, but another common rifle used in World War II by the Soviets that is often overlooked is the Tokarev SVT-40, also chambered for 7.62x54r. In the 1930s, the United States had adopted the M1 Garand semi-automatic as its main rifle, and the Soviet Union realized that it needed to modernize its army as well and began finding a replacement semi-automatic rifle to succeed the Mosin Nagant, in use since the 1890s. The SVT-38 was developed by TOkarev and issued to Soviet forces in the Finnish War in 1940, but the 38 developed a reputation for being heavy, clunky, and unreliable. The upgraded SVT-40 was developed and nearly a million of the rifles were made and issued to Soviet troops. It was intended to replace the Mosin Nagant, but the Axis invasion in 1941 disrupted further production. For the sake of simplicity and to get rifles quickly to the front kine, the Soviet Union resorted to mass producing the Mosin Nagant as their main rifle. Still, over a million and a half units of the SVT-40 ended up seeing service in the war. In the 1990s many SVT-40s were sold on the American market as surplus rifles.
If you want a 5.56x45mm NATO semi-automatic rifle that is substantially shorter than either the AR-15 or the Ruger Mini-14, look no further than the IWI Tavor. This is a bullpup magazine located behind the trigger, making the rifle significantly shorter and more nimble for tight conditions. It also accepts AR-15 magazines, which are commonly available and further increases its overall versatility.