Note: I don't travel, so it's not based on personal experience! I'm also not working at borders.
Some answers above were very good, all depends whether you need visa or not based on the passport you use at the border check.
The concern here is more about the name change not matching your boarding pass, in case you show the passport without that change you'll have to justify having dual nationality (easy, just the other passport), and papers to prove the name change maybe, to confirm you are the same person.
You might also have to match the boarding pass with the correct passport, as it's read from the chipset.
See the information at the bottom of the passport (MRZ, Machine Readable Zone), it's the information the border is using to register and control your passport. if you use the wrong passport, you might have issue to print your boarding pass, or get on the plane.
So, for boarding you need to use the passport matching the name when you booked the travel with.
At arrival, you can show the passport you want and/or the one preventing you from having the need of a visa.
You might even have to prove you don't need a visa for your destination at the departure airport.
The best is to ask the embassy of the destination country. Though, some border staff might not always do their job as they should and will annoy you anyway, whatever the embassy told you.
You can also ask at the airport, they should be used to the cases.
Some countries have different laws and habits with last (family) name, when marrying, and people with dual citizenship also has that issue with Last name, not only with First names.
Some countries always keep their maiden name, some countries add both maiden and wed name on the passeport (usually in Europe), some countries (UK, and often east-european countries) will completely drop the maiden name, and they have to change their name on all official papers (even their kid's birth certificate, it's stating "the mother changed her name" etc.). AND some countries, it's the man who get the woman's name too !
So, I suppose border's officers are somewhat used to see different names on different passports. you just have to explain calmly !
Though, I've heard report from travelers saying that some Asian country borders didn't understand what "ep" (short for "spouse" in french) meant on their passport... so, it's not always easy.
Show the paper, explain the situation, all should be fine. You might have to keep the name change proof with you (paper, or on your phone, etc.).
(or even gender I guess whilst your at it...)
I did one today.
For the record, I work in France as Passport production service, so I see a lot of different cases.
And today, I got someone who had an official paper (birth certificate and decision from justice department) stating "This person will hereby be known as female" (even though that person didn't do any surgery and is still physically a man). She got a name change few months ago, but didn't have the right to be called a woman yet.
"Sex" mention on passport and identity card is a very bad tag and wrong data to use for identification.
It should be "gender" seeing France allows people having a different sex on their paper than their real current sex.
Or it should be ... nothing about sex or gender. just the name, birth date (and place even if that's also not important), a picture and that's all. We don't care if you are a man or women traveling in the plane... unless planes have safest seats for women in case of crash landing ? haha
Or different prices ? but that's probably me living in a free country who doesn't care about man and woman's differences. I guess it's a very different case in some countries or religious places etc. This is not the place to discuss that.
Impressive score on dual nationality as well as that is easier said than done as an adult these days.
I might have another opinion on that. I see a lot of people with dual (or more, even if it's rarer, but not as much as you think) nationality every day. But that's part of my job
But it's not that rare or difficult to get it. long process for sure, but not hard. In France, it's about 3 years long to get it from an adult asking for it. (not getting if from marriage with a French; which is a faster procedure)
Example of triple nationality :
British father, Italian mother, living in France and having a kid. the kid has both britsh and italian. The parents asks for French nationality because they lived here for years, kid has school in France, they all speak french and feel as French citizens. parents have both French and theirs, the kid get three of them.
Or a mixed couple traveling in USA and giving birth there... the kid is American by law at birth, but also has both their parent's nationality (as long as the birth is registered in each embassy for easiness).
yay, 20 xp for that answer 
