I'm just browsing here and probably not nearly qualified enough to answer this, but here it goes.

Spanish colonies in the Americas were mostly focused on extracting the riches of the area, things like sugar cane and gold. There was also a large focus on converting the natives to Catholicism as Habsburg Spain basically wanted to recruit more catholics into it's demesne during the times of the protestant reformation. The Spanish set up things like the Encomienda system which basically entitled Spaniards to own a certain amount of natives, on paper they were to show them the ways of the Catholic faith, teach them Spanish and in return take tribute from them, but in practice this was just essentially slavery. This system basically pushed race mixing in the native population as by law, mestizos (half indian, half european) couldn't fall under the Encomienda, so natives sought out to integrate with the Spaniards to escape slavery. Race mixing was much more common with the Spanish, during the conquest of Mexico, the conquistadors were given women by natives and they instantly took them up as their companions. Basically the Spanish wanted to quickly extract the resources they discovered, and save (in their eyes) the local populace, which lead to the unique mestizo culture in Latin America today.

English colonies were a bit different, most people might try and say they were nicer than the Spanish (quick extraction of resources, enslavement etc.) but what it mostly comes down to is the fact there wasn't actually much in the way of resources in North America. There was no Aztec or Incan emperors clad in gold to conquer, and there wasn't a lot of the resources that was profitable for the Spanish (and Portuguese) in the new world like sugar cane. Settlers of Jamestown (the first successful English colony) brought sugar cane there and couldn't get it to grow. Now this led the English to basically build long lasting, sustaining communities, focused on agriculture and industry. This is probably one of the key factors as to why North America is more developed as a whole when compared to South America. The English mostly forced natives off their lands, and didn't try to integrate them into their realm the same way the Spanish did, however they were still cruel and justified their cruelty with religious reasons.

As for the French, I really don't know much about them. I've heard that French colonies were never really intended as permanent residences, just a way to get rich off things like fur and fish. The French are somewhat known for having better relations with the natives, they weren't interested in land so didn't have as many problems with them, and for whatever reason never really sought out to convert the local populace to christianity.

I'd like to add onto to Tepeyollotl's post, because I know a bit about the French colonies.

The French intermingled with the Indians in a way neither the Spanish nor English did. While the situation at Montreal and Quebec were somewhat different, the frontier settlements were very much integrated, if you'll excuse the anachronism. There is an important ceremony which characterizes much of their relationships, conducted by a missionary named St. Lusson at the Sault in 1671. The ceremony itself is about what you'd expect it to be, but it produces powerful fruit - the translator referred to the French king and his deputees with a term Onontio, which means father, which the Indians used to refer to French rulers well into English occupation. Michael Wittigen wrote a very interesting article called "Rituals of Possession" on the way which this ceremony and other French practices cemented a connection with the Indians. In short, many tribes did accept French rulers as onontio, although not in the French sense, but an Indian sense - a leader, a mediator, a gift-giver, and a wielder of sacred power.

I think the most important community to talk about when discussing their relationships is Mackinac, a trading fort at the straits between Lake Superior and Lake Michigan. Sleeper-Smith has done some great articles on the most important people of all there - Indian women. French traders found themselves at an impasse with Indians, who wouldn't trade with just anyone. Kinship, even vague relationships like babysitters, was absolutely necessary to engage in trade. So the traders would marry into Indian families - frequently, 'marry' just meant shack up with an Indian woman for a time and then leave. This was not actually a problem for those women, who found themselves very powerful agents in the fur trade; they kinship network then included half-Indian children (called meti), the French they could engage with rituals of Catholicism and the French language, and their expansive Indian family. Repeated marriages by these trader women were also quite common. The Jesuits were scandalized by the behavior of the traders and complain repeatedly in the Relation (an important set of letters and documents from New France) that the men are licentious, sleeping around with Indian women, drinking excessively with Indian men, and giving too many gifts to the tribes.

The town at Michillimackinac was an extremely vital link in the fur trade, and yet archaeological evidence (Walthal - French Colonial Archeology) suggests it was barely even palisaded and only a very small garrison ever lived there. This fluid mixture of economy and intermarriage went very well for them.

It went very poorly for the English, however, who racialized the Indians to an extent. Sleeper-Smith's "Unpleasant Transaction" article and others about the transition period paint the same story - where the French embraced the Indians as French citizens and the meti as of French blood, the English called them all Indians and were mistrustful of them - which they were right to be, given how they repeatedly broke promises to their Indian military allies. At some point Michillimackinac was evacuated for fear of the large meti community there, and the English built an extensive fort on the island in the straits. They were right to fear the mistakes of the Canadian government, of course - at one point, Indians convinced the garrison to leave the fort to play lacrosse with them, and as soon as they did, the Indians stormed Mackinac and captured it easily.

The essential and ridiculously influential book on French-Indian relations is Richard White's Middle Ground.

The French developed the best relations with the Indians as they needed their numbers to keep the British in check. They never had anywhere near the numbers the English had as they were not interested in settlements as much as trading posts. I've seen estimates that the French settlers outnumbered the English by 10 to 1. The French used the Indians to help get a major foothold and secure the Ohio River valley before the English got interested. They also worked together to harass the English colonies at the frontier to help check their expansion. The French essentially thought that the Indians were to be their trading partners alone, despite signing treaties stating otherwise.

The English explicitly wanted to settle the new lands and exploit it as best as possible. They used their model of Ireland to set up plantations all over the colonies to grown item, catch and salt fish for export to England, trade for furs and other goods and get just about anything they could out of the colonies for their benefit. They supplied large numbers of settles by promising land and also great numbers of indentured servants. Because they wanted to settle the land the Indians had to give it up.

The English had a roller coaster relationship with the Indian tribes. Some colonies made peace, some made war. A major event in the colonial history of English/Indian relations was Bacon's Rebellion of 1676 in Virginia and Albemarle (North Carolina), which started over tensions between colonist and Indian tribes. Bacon was famous for stating that you can't tell a friendly Indian from a non-friendly one so in his mind the only good Indian was a dead Indian. And he acted on it. It was the first major uprising in the colonies 100 years before the founding fathers got into the act.

You can read more about Bacon's Rebellion and Virginia's Indian relations in Conceived in Liberty, vol 1,Volume 1, Chapter 10. Relations with the Indians. It really was complex.

I am not big on Spanish colonial history but if I had to ball park it I would say they wanted everything they could get like the English, so if the Indians got in the way they had to be removed.