I would love to give a really detailed answer to this for you, but I'm at work so I don't have my books! However, I can offer you a few bits and pieces that might expand your understanding of the legions and their duties a little.

While some Republican soldiers fought a lot (like Scipio's or Caesar's veterans), most fought less often as in the Republic even armies that remained on campaign for years would not have seen many pitched battles, which were not the norm even on an active campaign. Most of the work of soldiers on campaign was labour, with marching, building, law enforcement, training, more marching and digging and building. In very violent periods, such as during Hannibal's occupation of southern Italy or the Civil War period after the death of Caesar, it would not be unusual for soldiers to confront other armies and perhaps even fight pitched battles. But I can't say this is the norm, as small-scale warfare like the 200-year subjugation of Iberia, or general 'peacekeeping' in places like Syria was more the sort of stuff most soldiers did on foreign posting, though consular armies did have a tendency to go looking for fights. To elucidate the experience of the 'average' soldier even in the Republic is probably to generalise too far, so I'll just pick a period for you.

Let's narrow it down the Principate (since that is when the 25 year service period was set, though it changed several times), which is the period of Roman history we all see in the movies. I'm talking between the creation of the Principate in the 20s BC and the outbreak of major, more regular warfare along the Rhine and Danube, and against the resurgent Persians in the east, in the mid to late 200s AD. This is probably the most 'peaceful' period of Roman history, but it was only really peaceful in the sense that there was little civil strife and the 'central' Empire, that is to say mainly Italy, was pretty much free from war.

However, from the reign of Augustus to the outbreak of the Marcomannic Wars under Marcus Aurelius (160s AD), there were multiple major wars. Tiberius had to handle the Great Illyrian Revolt from 6-9 AD, which saw eight legions in action. Varus lost three in Germania. Germanicus took eight legions to chasten Arminius in 14 AD. Agricola campaigned to all the way into Caledonia years after Suetonius Paullinus put down a major revolt by Boudicca around 60 AD. Corbulo campaigned several times against the Armenians and Parthians under Nero. Judea erupted in 66 AD and Titus destroyed Jerusalem in 69 AD, killing hundreds of thousands and finished the war his father Vespasian had been sent to first after Cestius Gallus lost a legion at Beth Horon. Trajan conducted multiple wars in Dacia and Pannonia and Mesopotamia. There are several other major wars beyond these.

So as you can see, in a long career during this peaceful period, there was a more than fair chance that you would end up on a campaign at some stage, and all of the campaigns I mentioned above featured at least one major battle. However, there was a lot more to being a legionary than fighting or just waiting to fight. The Imperial Legions were as much law enforcement and imperial bureaucracy as they were armies. They did policing duties, guard duties, frontier patrols, arrests of fugitives, the carried messages from place to place, they acquired food, fodder and other items for their camp or army which might see them away from camp for days, they fulfilled bureaucratic functions like processing complaints from locals, they were engineers and builders and built entire towns, they managed the postal service, and of course they trained. With all the duties Roman legionaries had to perform, they could be perfectly busy with no enemies in sight.

In summary, the amount of battle a soldier experienced depended a lot on where he was posted and in what years. Most legionaries if not all would have been on campaigns during a full career, though this did not necessarily entail fighting. Some on the other hand might have fought multiple battles or participated in enormous operations like the Siege of Jerusalem. However, I think perhaps the most important thing to remember is that the legions were not like a modern army: they fulfilled far more functions that are today fulfilled by police forces and civil services. So even if a soldier didn't spend much time fighting, it's not as if he never had anything to do. He could be running messages, surveying a new town, chasing bandits or patrolling frontiers and he was still doing his job.

EDIT: This post by /u/Iguana_on_a_stick is also very informative on this matter.

Here's a similar question posed about two years ago: https://www.reddit.com/r/AskHistorians/comments/23eqv4/how_much_combat_would_the_average_roman_soldier/

Edit: The askhistorians bot wants me to give proper credit, so, proper credit goes to [deleted]

I answered a related question last week, which talks about the things soldiers did when they were not fighting. It'll make a nice addition to u/PapiriusCursor's post, I think.

Spoiler: many of them were not just sitting around when they were not fighting.