Relationship of the texts
To determine how to reconcile the Gospels accounts, we first need to determine the relationship between the different accounts. The best way to do this is to look at the original Greek.
Below are the verses in question. Since a few Biblical scholars also think Luke 10:4 is related, I went ahead and included it too. The Greek is from NA27 (no textual variants significant to the discussion). The English is ESV. Words shared by Mark & Matthew are purple; Mark & Luke red; Matthew & Luke green; and all three blue. The word for staff is in bold instead of blue.
One common way to resolve differences between the Synopic Gospels is the Two Source Hypothesis. In this hypothesis Matthew and Luke are made from a combination of Mark and an unattested source called "Q", possibly with some additional sources thrown in. A few scholars have suggested that this hypothesis applies to our passages, and that Matthew and Luke were an attempt to combine conflicting accounts between Q and Mark in our passage
We don't have to take their opinion as fact, though. We can test it ourselves using the data. If this hypothesis was correct, we would expect to see a good amount of of green, as these are words which theoretically came from Q. However, there is a total of one word - ὑποδήματα ("sandals"; Mark's word for sandals is σανδάλια). Furthermore, the word isn't even negated in the same way between the two passages. All four passages share ὁδόν ("bag"). And that's it. Luke 10:4 has a unique verb, a unique idea (greet no one), and the two shared words aren't part of a phrase. Whatever Luke's source for 10:4, there is nothing but one's imagination to connect it to Q.
If we decided two sentences showed a literary relationship simply because they happened to share two nouns, we'd never get anywhere. Any conclusion that a relation exists here must be based on preconceived notions about the Gospel's relationship, as the text does not support it. If we do no decide in advance that every similarity between the Synopics must derive from Mark and/or Q, we will reach a very different conclusion about this passage.
Actual relationship
Now, of course, it is nearly universally acknowledged that copying did occur between the Synoptic accounts. This practice was common and considered ethical in the ancient world, and need not impact ones view of inspiration.
Here, the three parallel passages share a clear theme. However, the grammar does not provide evidence of copying. There are some shared words, sure, but much less so than in parallel accounts usually do. Furthermore, even where there are shared words, there are differences in word order. Mark states Jesus made a command; Matthew is just the command; and Luke write "He said" - three different ways to convey the idea of Jesus issuing a command. There are never more than three shared words in a row.
If one is committed to an idea that two of the accounts must have been copied and adjusted from the others (and possibly combined with an outside source), one will struggle to explain the great grammatical differences. It takes a lot of hoops jumping to explain why two author drastically changed the grammar, word order, etc., while leaving the basic theme unchanged.
If, however, one is open to the possibility that the accounts are independent of each other, the solution is simple. What we have here is not dependence, but coincidence. The parallel accounts represent three different eyewitness account (in this case, not universally throughout the Gospels). This hypothesis best explains the actual data. Additionally, it makes sense that such an important event for the Twelve would have multiple remembrances with eleven possible witnesses telling the story. The fact that discarding the shortcut of using the Two Source Hypothesis to explain every difference means we have to treat each passage individually is not sufficient reason to use it where it does not actually fit the data.
Contradiction resolved
If the accounts are indeed independent, in my opinion there is no need to resolve the differences. I believe that the minor difference in details between the Gospels actually are excellent evidence of their accuracy. Real, legitimate eyewitness testimony always have such differences. When they don't exist, it means collusion took place, as any lawyer will happily point out in court. Differences, however, don't imply someone is wrong (or lying) - it just means two people saw the same event from a different perspective. That said, since the OP requested a resolution, I will provide one, or actually two.
The first option is quite simple: the accounts don't refer to the same occasion. While it is natural to think of the Gospel accounts as giving the entire story of Jesus' ministry, that is not really the case. It would be foolish to say Jesus never gave the same speeches\taught the same lessons twice just because they appear in the Gospel only once. Likewise, there is no real reason to say Jesus didn't send out the 12 on multiple occasions.
However, there isn't even a need to postulate multiple mission trips. The key to seeing why is in the verb. Matthew uses κτήσησθε while Mark and Luke use forms of αἴρω. According to BDAG, the primary meaning of κτήσησθε is "to gain possession of" suggesting a gloss along to lines of "procure" or "acquire". The semantic range of αἴρω is quite broad. It can mean "to raise to a high place" (pick up), "to life up and move from one place to another" (take/carry), "to take away, remove, or seize control of" (take away), and a couple other more technical or metaphorical uses.
Jesus, of course, wasn't speaking Greek when he dispatched the Twelve. Thus, (assuming Jesus did speak the words), the accounts reflect a translation of the original Aramaic - and in any case is not word-for-word. As anyone with any translation experience will realize, there also is not a one-to-one correspondence between languages.
Mark is capturing the idea of Jesus saying (paraphrased) "take what you have on hand (a staff and sandals) and go". Matthew and Luke are capturing the same idea, but as "don't go and retrieve anything (e.g. spare sandals), but go right away". They both mean the same thing - go at once. This explains Mark's αἴρωσιν ("don't carry anything but") and Matthew's κτήσησθε ("don't acquire anything at all"). That just leaves Luke's αἴρετε.
When doing translation/analysis, we like to think that the same word always means the same thing. But, the reality is that words 1) have multiple meanings and 2) don't mean exactly the same thing to every speaker. When one looks at Luke's other uses of κτάομαι (Luke 18:12, 21:29; Acts 1:18, 8:20, 22:28), one gets the strong impression that to him the word had a narrower meaning of, or at least a strong connotation of, "purchase". Since he wished to capture a broader sense of "don't acquire (go and get) anything", he chose a broader word - αἴρετε. As further evidence, for Mark and Luke having different word meanings in mind consider:
Whoever wants to be my disciple must deny themselves and take up (ἀράτω) their cross and follow me. (Mark 8:34)
And whoever does not carry (βαστάζει) their cross and follow me cannot be my disciple. (Luke 14:27)
It seems when Luke wants to say "carry", he uses βαστάζω and when Mark does, he is OK with saying αἴρω. The semantic range of βαστάζω, αἴρω, and κτάομαι overlap in general. The data suggests that Luke has a slightly different understanding (or preferred usage) of αἴρω than Mark and a slightly different understanding of κτάομαι than Matthew.
Thus, all three accounts say the same thing - leave immediately, taking with out only what you have on hand. Whether the texts refer to the same event or multiple events, there is no contradiction. Instead, the textual evidence suggests we have three independent eyewitness accounts.
