As you've been told, you should be able to do your own research and decide by yourself what is the best android phone for you, because we don't know which specifications you care about.
Anyway, now, by the end of the year, it's a perfect moment to review new phones and decide which one we prefer, because all phones are already presented and on sale. So I will try to make a complete list of available phones, you can compare them and find out what do you want to have.
I will say the name of the phone, link you to a web when you can look up about all the specifications, and try to resume the important info about that phone. I won't mention phablets since you said 5.7" seems too much for your likes, and will try to avoid personal opinions in this part.
1. Samsung Galaxy S6 (Full specs: http://www.gsmarena.com/samsung_galaxy_s6-6849.php)It has a great camera, probably the best one in the market. You can watch a lot of videos comparing the different top-android cameras anyways. The phone is very fast and effective, if you come from an iPhone 4S, you will really notice this is a pretty good and fast machine.
It has a very good screen too, and I'd like to say that It's really difficult to make scratches on it (It is made of Gorilla Glass 4).
Despite of that, we have to say that the battery life is pretty bad. You probably can't survive an entire day without charging it. (At least, it can use Quick charge, so you will be able to charge the 100% of the battery in about 72 minutes (In fact, Samsung says 72 minutes, but if you look for more info, each review says a different timing... so, I can't be sure about this, but I can say the felling is that it's much more faster than the normal way of charging)
It doesn't have a MicroSD slot. (There are 32gb, 64gb and 128gb models, but it will get potentially more expensive)
The phone costs about 579 euros here in Europe, and 699$ in Amazon.com (probably you can find it a bit cheaper, I just don't know where to look for the US price...
)
2. Sony Xperia Z5 (Full specs: http://www.gsmarena.com/sony_xperia_z5-7534.php)Again, a great camera, maybe the best one. (I prefer the S6 one because it's faster. But like I've already said, you can (and should) look for video reviews of the cameras to get your own conclusions). The phone is very fast too (I think you won't notice any difference between all these phone's fluency since I can't notice it, but some people say they do... normally in favor of the phone they have, but well...
)
This phone is waterproof. And though there have been some problems about this (a lot of drowned phones), if you take this function more carefully (don't go scuba diving while texting), it can save you from some fails you can have in your routine, like dropping the phone in the bathroom or being in the street when it starts raining.
Moreover, we have to underline the battery in this phone, because it is really good. If you are a normal user, you will probably charge your phone once every two days. And you can use Quick charge too.
The fingerprint maybe it's bad set for left-handed people since it's designed to be comfortable when you unlock your phone with your right thumb, and the screen is not in 2K resolution. (It's one of the best screens anyway, the colors look amazing in it.)
You can buy this phone for 699 euros, and for 634$ on Amazon.com (I think I have just found a bargain boy!
http://www.amazon.com/Sony-Xperia-E6653-International-Stock/dp/B015ZLA37I/ref=sr_1_3?s=wireless&ie=UTF8&qid=1446490265&sr=1-3&keywords=sony+xperia+z5)
If you are interested in get one of this top-phones, I have to say that those two are the best for me. But, again, you can watch videos and read 'pro' reviews to get your own opinion. You can check
LG G4 too. Other options like the new Nexus 6, the new LG V10 or the Galaxy Notes from Samsung, are 5.7" or bigger so I'm not talking about them.
If not, I will give you some other examples of good phones with a more accessible price.
3. OnePlus Two (full specs: http://www.gsmarena.com/oneplus_2-6902.php)Called the flagship killer, this phone has really good specs with a lower price than the rest. It has a good camera (not as good as the one in the Z5 or in the S6, but pretty good anyway!), a good battery life (it will survive all day without charging, and it counts with the Quick charge too) and It's a great phone if we talk about its speed and its fluency (It has the same chipset as the Z5 (Spandragon 810) and 4GB RAM, 1GB more than Z5 and S6, so you can be sure this is real flagship killer.)
I have to warn you that it's difficul to get this phone since you need an invitation for it. But don't panic. If you really want it, you will always be able to buy it from some chinese pages. If you do this, you won't have the guarantee that the OnePlus team offers.
Some more important info about this phone is that it uses a USB Type C to charge, so you can't use your old Micro USB cables to charge it (you don't care about this if you currently has an iPhone anyway.), and I have to say that the screen is definitely not as good as the one we find in the Z5 or in the S6.
It doesn't have Micro SD slot, so probably you will need to purchase the 32 gb model (there are two models, 16gb and 32gb. The 16gb has 'only' 3GB RAM)
If you get an invitation, you can buy it for about 400$ (the 32GB model). If you buy it in another webpage, probably it gets a bit more expensive, but not that much (I suppose it will cost between 20 and 40 more dollars)
For similar prices, (around 400$) you can find some other good phones, but I will just mention them to you because they are really similar and I don't want to make this endless.
5. Meizu MX5 (full specs: http://www.gsmarena.com/meizu_mx5-7240.php) (About 370$ on Amazon.com)
6. LG G3 (full specs: http://www.gsmarena.com/lg_g3-6294.php) (About 270$ on Amazon.com. Very good price.)
If you still want something cheaper, keep calm, you can look at this one:
7. Xiaomi Redmi Note 2 (full specs:http://www.gsmarena.com/xiaomi_redmi_note_2-6992.php ) (About 160$ on Amazon.com)
To be honest, I think the gap between the top-phones and the good mid-phones is broken. The difference between that Xiaomi Redmi Note 2 and the Samsung Galaxy S6 is definitely not enough taking into account that the Galaxy S6 is more or less five times more expensive than the other one (crazy
) But if you have the money, I bet you'll be really happy with one of the super expensive phones.
To end with this, I want to say that I've made all this by my own research so It has been written following my opinions in some parts. I invite you to take this to have a first idea of what you can buy, and make your own research about each phone, and all the phones I didn't mention here. I want to say too that the LG G3 price has surprised me, and I want to recommend it to you if you want a phone for about 300$. If you don't mind about money, I will take the Z5 since the other options for that price are very big phones. I'm using a Z3 now and it's the best phone I've had, so I would trust Sony flagships.
I had and iPhone 4S some years ago, and I can tell you you are doing a great move choosing an Android, you will love it!
Btw, I read the FAQ section and I think I'm not doing anything wrong with this long message, but sorry if I broke any rule guys.
