I am a resident of Turkey and I will go to Austria for 6 days beginning of June and then I will come back to Turkey. After staying 10 days I will go to Italy for 11 days at the end of June. Do I need two single-entry visa, or a multiple-entry visa, and which country do I apply to? (My first destination is Austria but I will stay more in Italy)
The rules have changed in 2020, see Schengen visa(s) for two events close in time for more details
The issue comes up often and unfortunately, there is no perfect solution for this scenario. A multiple-entry visa would of course be a good one but those are mostly intended for trusted travellers who have been to the Schengen area before and have a reason to visit the same country repeatedly. You can always ask but you can't really make sure you will get one.
If you were planning a single trip with several destinations, you should apply to the country that will be your main destination (e.g. where you spend the most time), the regulation is very clear on that. But in this case, you are really doing two distinct trips and there are no guarantees that either consulate would be willing to issue a visa covering both.
If you have the time and money for it, you could also spend the time between June 12 and June 23 in either Austria or Italy. You would then have a single trip with a clear main destination and could apply for a single-entry visa accordingly.
For two distinct trips, the most straightforward approach would be to apply for a visa for the first trip, use it, come back to your country of residence and only then apply for a visa for the second trip. If you still have all the documents from the first application and you can show that you used the first visa correctly (i.e. you didn't cheat), getting another one should be very easy. But 10 days is a short time to get a visa (although if everything else fails, it's not necessarily impossible so keep that as a back-up solution).
You could also apply for both visas in advance (in April-May), starting with Austria and then Italy. My understanding is that as long as they don't overlap, it should be possible for the Italian consulate to issue a visa even before you used your Austrian visa. You would need to pay twice of course but you would otherwise be certain to be able travel as you wish.
What you need to avoid is getting something like a single-entry Austrian visa valid for the whole month of June because it would not be enough for your trips but would prevent you from applying for another visa from Italy before using the Austrian one.
Another solution in your case would be applying to Italy first, as soon as possible. If you are lucky, they might issue a multiple-entry or two-entry visa valid for June-July, and you are set. If they only give you a single-entry visa valid from June 23rd, you could still apply for another single-entry visa from Austria to cover the first trip. And if they decline to process the application, you still have time to lodge another one with either Italy or Austria and you have lost nothing (in the case, the consulate should refund the application fee and return the documents).
Whatever you do, you can always write a cover letter explaining your situation and join it to the application. Hopefully it will help the consular officer understand the problem and issue what you need. Also make sure to double-check all the documents and to have a very strong application. If you have everything ready, it will be easy to resubmit it to the other consulate if needed and ensure it will processed in a timely manner.
Finally, if you get an Italian visa, be aware that the Austrian border guards might ask you about it. If they ask, simply explain why you did it this way and have all documents pertaining to your Italian trip with you, if they want to see them.
The EU's Handbook for the processing of visa applications and the modification of issued visas has this to say about such a situation:
Recommended best practice: In case an applicant is to travel to several Member States on different trips within a short timeframe, consulates should apply a flexible approach and consider this as one trip for the application of Article 5(1) of the Visa Code to avoid unnecessary burden and costs for the visa applicant.
Example: An Indian student residing in London (United Kingdom) wishes to travel to Denmark (15-18 August 2014 (4 days)) and to Spain 3-12 September 2014 (10 days)).
In this case the Spanish consulate should deal with the application and the visa issued should cover entries into and stays in the two Member States.
Since this is a recommendation rather than a rule, you cannot be sure that a consulate will follow it, but it certainly makes it worth a try.
The case in this question seems is closely analogous to the example for the handbook, so the asker's first plan should be to apply to Italy for a visa that covers his combined trip. Note that combining the trips for the purpose of the visa application means that both of them must be documented in the visa application.
Beware that there are versions of the Handbook floating around that don't contain this recommendation. It is there in the Consolidated 9 July 2014 version (hosted from a Danish government site) but not in the 15.9.2011 version that Google finds hosted on the European Commission's website). The fact that it's the newer version that has the recommendation seems to be a good sign; actual consular officers can be expected to work from the newest version.