There really isn't a significant interchange of ideas between Muslim and Christian communities in the Middle Ages. Christian concern with Muslim knowledge was mainly focused on a refutation of Muslim religious beliefs and a defense of Christianity. For example, people have pointed to Peter the Venerable's translation of the Quran in the 1140s as a sign of inter-religious dialogue, but really, he just wanted it so that he could refute Islam.
The real advances in Western 'scientific' understanding do nonetheless come from the Muslim world, but they do so indirectly by way of Byzantium. It is the two great influxes of Greek knowledge, first in 1204 with the Fourth Crusade, and then in 1453 with the fall of Constantinople, which are the real source of most of the texts in the west, not the Crusades to the Levant, interactions within communities in Spain, or the Reconquista.
For more on Christian-Muslim interactions and Christian understandings of Islam, see:
I don't think we can be quite so dismissive of Western attitudes to Islamic authors as telkanuru has suggested (although I may be misunderstanding the thrust of his/her point). Though their point seems to relate to Western relations Muslims qua Muslims, in which case the interest was certainly religious and focused around the inter-religious polemics of the twelfth and early thirteenth centuries in the west. And indeed they may very well be correct that the majority of texts came from byzantium (I am not up to date on the scholarship regarding this particular issue). It is worth noting some significant points of interaction.
On the side of science and the arts, there was certainly important relations between the Latin and Muslim worlds (although I think it would be slightly less misleading if we said Arabic, rather than Muslim) besides the intermediary of Byzantium. As idjet notes, quite a lot of texts did indeed come from Toledo. Though we can start our story earlier than this.
We can take, for example, the case of Constantinus Africanus, who came from North Africa into Italy in the mid eleventh century. He became a benedictine monk at Monte Cassino and translated into Latin a vast quantity of Arabic medical texts.
This sort of story is still the case in the twelfth century, however Latin authors have at this point begun to take on the work of translation themselves. So to take telkanuru's example of Peter the Venerable's translation of the Qur'an, he went to a man named Robert of Ketton to translate it. Now the reason that Robert of Ketton was an experience translator is that he had already made a name for himself translating scientific works from Arabic into Latin. Similarly, even on the topic of his Lex Mahumet pseudoprophete, we first of all know that a Muslim collaborated in the process of this translation (a certain Moor Muhammad), but we also have good reason to believe that in the course of this translation, significant Islamic commentaries on the Qur'an were consulted in the translation of difficult passages.
Finally, just to round out the dialectical side of the picture, we can simply enough note Aquinas not only referring frequently to Ibn Rusd (Averroes) but as "the commentator", given his significant influence on western interpretations of Aristotle. Avicenna's works were also influential on western authors in from the late twelfth century onwards. (Edit: It is also worth noting on this point that Gerard of Cremona's translation of Avicanna's Canon of Medicine remained one of the standard medical texts in the Latin world until at least the fifteenth century.)
So to summarize, my point here is not to give an account of precisely how important Arabic thought was to the Latin world. Rather I am merely noting that they were not an insignificant source of both texts and knowledge that had a noteworthy impact on the Latin world (as evidenced by a few famous examples). Finally, I would like to reiterate idjet's point that we shouldn't fall into the trap of viewing the development of western science as merely the result of interacting with x or y library. Rather it developed out of the academic dialectic that developed in the high to late middle ages in Western Europe (in which these interactions with Greek and Islamic libraries and scholars certainly played a notable role).
edit: a bit of proofreading and editing for clarity
Maybe I can add something to the "art-part" of the question. I'm not sure if most of the population of christian Europe had much knowlege about the arts of the islamic world. But in certain contact zones like Spain, southern Italy or Byzantium there was quite a vivid exchange. Muslim palatial architecture for example was copied by several christian rulers around the Mediterranean (Alcázar of Seville, Zisa in Palermo). Even church interriors could be influenced by islamic art (Capella Palatina). Smaller works of art from muslim craftsmen found their way into many European treasure chambers, even far away from their original place of origin. The famous Murano glass from Venice was highly influenced by islamic glass production. In the case of many pieces of art that were produced around the medieval Mediterranean it is almost impossible to properly assign a christian or islamic origin to them because they shared a basic artistic language.
The thinking types generally engaged with it. I have no anecdotes of personal opinions on Islamic sciences, but I do know that even by the 13th century, as the Christians advanced through Spain and encountered well-stocked Muslim libraries full of advanced work in Science, Philosophy (they uncovered many works of Aristotle previously unknown or thought to be lost, for example), Astrology, and Medicine, their immediate reaction was to engage it. Especially philosophy: the works of Averroes and Maimonades (Jewish), for example, would become highly debated by Christian thinkers during the early development of Scholasticism. The works of Averroes were condemned by a lot of people...but this condemnation was the result of philosophical debate. They declared Averroes wrong, not invalid; and their criticisms echoed earlier Islamic scholars like Al-Ghazali, whom Averroese argued against.
A more engaging example would probably be Jewish medicine, which came of age within the Islamic world. While perhaps higher churchmen, occasional Popes, etc., would come out and declare that 'no Christian should see a Jewish doctor', a lot of (important) people still knew that Jewish doctors were the best doctors around. The kings of Spain were known to keep Jewish doctors in their retinue, and notarial registers are full of Christians paying Jewish doctors for their services.
So, in spite of our best efforts to believe otherwise, medieval people were often able to see the forest for the trees. The fact that some thinkers were Jewish or Muslim didn't mean that they were dismissed or condemned outright for their thoughts on things outside the theological realm.
Most comments seem to focus on western europe, relegating the eastern part as an intermediary. Can anyone give an answer on how the situation looked like from eastern europe's perspective instead?