First and foremost I must recommend The French Foreign Legion by Douglas Porch, it is an exhaustive, and in english at least, definitive history of the legion. I dont have my copy on hand, but when I get home I can give exact quotes to support this summary.

The Legion was inherently very different from the regular, conscript army even from its beginning in 1830, it had a much more professional, mercenary feel. It was created out of expediency as in 1830 France was an exile refuge of revolutionaries throughout europe, and the July Monarchy wanted to ease tensions by essentially shipping them out to the new colony of Algeria, after briefly fighting in Spain against Carlist revolutionaries. Algeria became their home and base, becoming one of the three main forces (along with the Zouaves and the penal Battaillons d'Afrique, though after 1871 these essentially disappeared) in charge of its "pacification". They played a large role in the failed Mexican expedition, and it is there that one of its central regimental legends, the last stand at Camerone where they fought to the last man, was cemented. They were present during the Italian war against Austria and played a key role in the Battle of Magenta. Shortly thereafter they participated in the defense of France in 1870 and the crackdown of the Paris Commune in 1871

1871-1914 was basically the Golden age for the legion, where they and the French Marines formed the spearhead of colonial efforts of the period. They led the forces in Dahomey, Indochina, Madagascar, Syria, and Morocco. They were pretty much THE unit in charge of colonial ventures. Furthermore, commanders who cut their teeth in the legion would go on to become prominent generals in the metropole army.

In WW1 they did fight in france (though they left austro-german enlistees in Morocco for obvious reasons) and played key roles in scattered offensives, mostly in 1914 and 1917.

For the interwar and WW2 periods again they were relegated mostly to the colonies. However, they once again became key units in the decolonization conflicts, forming the backbone of french forces in Vietnam and Algeria.

In fact, to this day, the 3e Régiment Étranger is one of the most decorated units in the entire French military.

I understand this is heavily summarized and I can go into much greater detail on recruitment, national makeup of the Legion, and combat doctrine when I get back home

Always more to be said on the topic, but you'll likely find this brief history of the Foreign Legion I wrote awhile back to be of interest!

All right, expanding on my original response I can go into more detail. You said you were more interested in the earlier history of the legion. Douglas Porch in his work shows the semi-conscious connection between the Legion and the Ancien Régime's reliance on paid, foreign-national troops. The Revolution(s) changed this attitude, by 1830 these units were seen as "mercenaries" willing to fight for any cause, and a recruitment bill passed in 1832 stated that "no one will be admitted to serve in the French forces if he is not French." This attitude meant that for a long time the Legion was seen as something approaching a penal battalion; necessary to get rid of unsavory types from France, but not a unit of any military worth or virtue. Then Minister of War Soult wrote in 1834 that "The government has no desire to look for recruits for this Legion. This corps is simply an asylum for misfortune."

Thus the Legion, after being rented out to the Spanish Crown for a bit, was purposefully resettled to Algeria. It wasn't until Marshal Bugeaud was appointed commander-in-chief of Algeria in the 1840's that it would come into its own as a crack colonial force. He built the Legion into a unit focused nearly exclusively on harsh-but-fair discipline, and peak physical endurance: the lack of cavalry meant that to subdue Algerian rebels, Legion columns had to regularly undertake quick, hellishly long marches under blistering heat. This also meant that for a long time, Legion style combat differed vastly from European doctrine, with a focus on mobility and utilizing cover rather than drill formations.

As for recruitment, until the globalization of the mid-late 20th century, the main source of men for the Legion were Germans, followed by Swiss, Belgians, Poles, Spaniards, and a smattering of English and Italians. Before WWI, french citizens made up only roughly 40% of the Legion, and this percentage was far lower in the years before 1871. The romanticization of these men being scorned "bourgeois" lovers or scoundrels fighting for a cause, such as in Beau Geste, of the Legionnaire as a patriot or at least man of conviction was mostly a myth. Until 1914, most Legionnaires were older men with a background in skills and trades, and typically middling to lower class. Regimental pride and a sort of professional sense of duty were the main motivators. It certainly wasn't the pay, so ridiculously low, the sale of bits of uniform or equipment was regularly and harshly punished. Desertion was always higher in the Legion, facilitated by the anonimat (for a long time, recruits had to give almost no personal information on enlistment, and many could and did enlist under assumed identities).

I have the book on hand if you have any questions!