Is there a central place where I can request my own IPv6 subnet ?
IANA is the central location, but only the five RIRs can get addresses from IANA. If you are big enough you can get addresses from one of the RIRs, which one of those is ready to give you addresses depends on where in the world you are. If you are too small to get addresses from a RIR, you need to get them elsewhere.
I usually request IPv4 networks from my local ISP, but is IPv6 handled the same ?
It is supposed to be handled exactly the same.
My ISP doesn't deliver IPv6 connectivity yet.
And that's where the problem begins. If you get IPv4 addresses from your ISP you should be getting IPv6 addresses from them as well. But your ISP will probably say they don't have customers for it, which is why they don't offer it (which BTW is why they don't have customers for it).
And since your ISP seems to be a decade behind schedule, you'll need to find an alternative. The possibilities depends on how large the network is, how urgent it is, and how much work you want to do setting it up.
The least amount of work you can do is wait for your ISP getting their act together and deploy IPv6. Several ISPs have promised to get going one month from now. You could check if your ISP is among them. If they are, then waiting for a month might be the way to go.
The second least amount of work you can do is to setup a tunnel through HE or another provider. This is a good option for many people, and it is probably the best option if you mainly want to learn about IPv6.
Next an option is to switch to another provider. But in that case I would recommend waiting until we see how things go a month from now before making a decision (unless you are in a hurry).
Another option is to set up multiple redundant tunnels through different providers such as HE and SixXS. Could offer more reliability if done right. But if you don't know how to do it right, you could easily end up in a worse situation than with a single tunnel.
If you have a really big network you could get addresses from a RIR and look into peering. HE has some sort of BGP tunnelling for that. I don't know the details about that, but it is certainly lots of work. And from the questions you are asking I guess your network is too small to consider this option.