You are 100% correct in your train of thought...
These days I am noticing (myself included) I think I know a lot more than I do or I am more capable than what I truely am because I have everything ready made for me. Crossing over from "There's an app for that." To I want to make an app for that and hardware for that. Requires a bit of research a lot of reading before one can even decide the approach they want to take.
I agree thingspeak may be a good way to start indeed...
However more importantly I would like to ask the original poster what it is he/she is trying to do remotely?
If you are just simply looking to access the device remotely then I will say the easiest way is using basic and a cheap but effective device called the nodemcu with a 4MB flash.
It will make development of any project you can think of easier, faster, and able even if you choose not to use basic mmiscool section of this forum the best package to learn from and develop with for beginners and seasoned professionals alike is the nodemcu. The best bang for your buck period.
Now if you simply what to access your device remotely with basic you can just forward the port as discussed and then just visit your IP a rather simple means from here you can access everything. Basic is entirely web based once it is flashed to the esp8266 v12 or nodemcu. Updates to firmware can be done over the air. You can write code using the editor in your browser. You can change the settings easily from your browser as well remotely or locally with very little effort really you could even just host an html page really but I'll warn you these chips are not made to serve up web pages and if there is too much html action you will overflow the buffer causing a reset. The thing is if remote access is what you crave then there is no simpler form then a nodemcu with basic interpreter firmware on it that I have found thus far. Java may be the other way or LUA. But for the beginner I'd suggest basic as it will allow you to easily do basic things like turn stuff off and on.
If you view the link in my signature it will get you started.
The fact is though you must share your goal in mind in order to know what it is you are trying to do so that you can get help it is obvious you want remote access to the device but what do you need the remote access for?
EDIT
SORRY ID DID NOT SEE IT WAS A SERVO!!! my mistake...
Mmiscool did a live servo broadcast at one time with the arduino and basic might want to look at that was pretty nifty... Which is just another reason to suggest basic.
Whatever you choose, Basic, Lua, Espirino, C based SDK's, etc... You need to define a goal and share your goal if you are looking for help to reach it. If you wish to hide your goal find a similar project that is not your project and use that goal to gather information to reach your goal.
For instance if you want to turn a light on and off why not say you want to turn an ac device on and off remotely.
At least give some instance of what it is you are trying to do.
These interpreters and sdk's etc all fair better or worse for different projects due to the way they are made but one thing is clear hardware resources are limited there are ways around this limitation by using other hardware to work with the esp8266 etc stuff like the arduino, a server such as a computer or even something as small as a raspberry pi your wifi router believe it or not actually allows your esp8266 to process more data then if you use the esp8266 and connect to it as an Access Point. Because your connection alone to the esp is gobbling up valuable resources that normally would not be used when connected to the wireless router.
All things to keep in mind.
I wish you the best on your journey.
I highly suggest if you are looking for remote access to anything you look up the fundamentals of networking such as configuring your router and so on...
Read the manual of your router no two routers are exactly alike to be honest. The best way to find out how to access your router is by reading the manual it should give you information normally 192.168.x.x where the x's are can be different depending on the make and model. From there you can set port fwd to 80 or DMZ the ip of your esp. Depending.
possibly start here http://www.smallnetbuilder.com/basics possibly if you want to look into some of the networking basics.
There is a difference between real fabric and shortcuts such as broadcom's ctf.ko we as consumers are getting ripped off by misnomers. We actually have routers that are barely capable of a 100MB wired connection but they use software to evade the NAT to allow for faster speeds which to me wreaks of security override which leads to vulnerabilities not healthy for networking at all really. Our hardware is not capable of speeds that exceed this without something that ignores the data processing that the NAT and firewall which is put there to correctly handle data inbound and out for security, integrity, and safety of your data . Essentially we are being sold one thing that claims it will handle 1GB wired data transfer when really there isn't enough fabric to provide this without taking shortcuts. All those high powered consumer routers are all culprits of shortcuts to real fabric.
Err sorry for the rant but it is amazing on how much as consumers we think we know but do not.
You do have enough info here to conquer your ideals of remote viewing or control but no matter how much we tell you it is going to take effort on your part. Using the examples of code given etc all important to understanding how it works as GK is telling you. Start doing some testing some experimenting so how the code works what makes it tick etc.. You will be amazed what you can learn then you say to yourself what if I change this code a little here or a little there then you will come to your own recipe with just the right amount of what you need.
Wish you the best people will help you along the way I am not saying don't ask questions but rather try to understand the question it is you are asking break it down into smaller more targeted questions. Now that you have a good place to start you can begin your journey into the understanding of and making things possible.
Another interesting project is Domoticz I believe you may bet some ideas from this as well.
Anyhow wishing you the best in your venture.
-forlotto
Where I buy my ESP8266 boards from... (Banggood)