Should I give my current employer a chance to counter the offer as a common courtesy?
Either you want to leave, or you want to stay. I don't think it's a good idea to give your current employer said chance (prior to resignation). I definitely don't think they'd view allowing them to counter the offer as courteous, more likely just an attempt to play them to get more money out of them.
If you want a counter offer, during the conversation to hand in your resignation, you could approach it slightly less directly (than mentioned below), sound less sure of making the change or mention reasons for leaving which are easy to change, which could increase the chance of getting a counter offer.
You shouldn't bring up a counter offer yourself (at least in my opinion) - you could list the reasons as wanting to move as primarily related to money and perks (and not say anything bad about them, and plenty of good things), which they could probably give a counter offer to match, if they want to.
Some advice had you been working there for longer, which may, to some extent, apply to you - if you were to consider a counter offer, I'd suggest having good reasons for leaving that could be changed (beyond just money) or nice perks at the target company which could be matched - while some people will go for the highest paying job in their domain, employers tend to like to feel that that's not the case - they want someone who likes working for them, likes what they do there specifically, likes the culture and the perks, and has a bit of loyalty - while honestly expressing loyalty while handing in your resignation is difficult, to say the least, you should be able to express the rest. If they were to bring up the possibility of a counter offer, you'd then negotiate making changes to address said reasons or perks. Then again - if you wanted to address these reasons (rather than playing employers against each other to get out as much money as possible), they might wonder why you didn't do so prior to handing in your resignation.
Given that I want to keep working for this company until I leave, when would be the best time to quit, ...
I recommend telling them as soon as possible. You're likely still largely in training (depending on the job, of course), so telling them that you're leaving in 6 weeks will probably allow them to maximize your usefulness during this time, rather than waste time training you (from their point of view, and possibly from yours as well).
But you'll need to check what your contract says regarding the amount of notice required (note that you may still be on probation, which may have a different time frame). In certain cases, you may even end up being escorted from the premises right after handing in your resignation. With the exception of this, I expect that employers will see you giving way more notice that what's required as a good thing, however, don't expect this to happen everywhere - I can't imagine it's common, but some might expect you to leave when the end of your notice period comes - if staying is more important than maintaining good relationships, hand in your resignation at exactly the required time.
Just make sure you have a signed contract at your destination employer prior to giving notice, as not to end up unemployed by resigning when your new employment is not legally confirmed and then withdrawn (it being 'legally confirmed' doesn't mean it's not trivially easy to get rid of you in many parts of the world, but I don't expect many employers to do so without very good reason, which is unlikely to have much to do with you personally, and I guess in some parts of the world 'legal confirmation' might not happen until your first day on the job, so this doesn't necessarily apply to everyone).
... and how should I do so in the most professional way possible?
Walk into your manager's office and ask them if they have a few minutes (or, if you work in an open plan, or he shares an office with a third party, ask if you can have a word in private - in a meeting room, presumably).
Take your resignation letter with you (you can find some templates for this online - be sure to check out a few of them, as a badly written one can leave a bad impression of you) - special mention - it should include the date that you're leaving, and the date you wrote it (as letters often do), for reference.
Say that you're resigning (and possibly include some apology about doing this so soon after starting), and let your manager take it from there - they should enquire about why you're leaving (stay away from reasons which might leave a bad opinion about you, if at all possible), where you're going, possibly how much you're going to get paid, possibly if there's any way they could keep you, etc. If they don't ask any questions, you can probably assume they're not really interested.
Then thank them for the opportunity of having been able to work for them (or do so prior to the questions).
You should probably spend much of your last day cleaning up your computer of any personal data (e.g. clear cookies and delete history in browser, delete personal files) or anything else, and saying goodbye to everyone (thanks your boss again) - if they require an exit interview, you to fill in any forms, or anything like that, they should tell you.