Inexperienced tea drinker could use some advice.

It might be better to get more accustomed to tea brewing before investing in a gaiwan. Brewing in a gaiwan is much more finicky, plus you have to be pretty into it. At that point the tea drinking is an experience, as opposed to enjoying a cuppa while going about your day. Both ways are perfectly respectable, of course. :)
Find out which greens and whites you like best, experiment more with brewing Western style, and then by all means get the gaiwan. You can definitely have a more exciting tasting experience with it. :) They can be pretty pricey too, so you want to make sure you really want it and think you`ll take the time to use it and save yourself from potential buyers remorse.

Tea vendors will usually have brewing instructions (how much to use, what temperature, steep times) for each of their teas either on their website and/or on the pouch. If it turns out they don’t, you can always look up tasting notes and see what other people did if you feel unsure of experimenting on your own. :)

For re-steeping, the trick is to steep it longer than the first time around to still get a flavorful cup. For example, if a tea was originally steeped at 3 minutes, I’ll do my second steep at 5 or 6 minutes. As for keeping the leaves, I tend to do a second steep pretty much immediately. Personally I wouldn’t let the leaves lay around for more than a day. If you leave it overnight, I’d put it in the fridge, but do keep in mind that with any length of time the constant moisture in the fridge can spoil it. It really is best to use them the same day though for the best flavor.

For temperature, some of us have an electric kettle that can boil at different temps. (For example, mine can go to 175, 185, 195, 200 or 212 at the press of a button) The rest guesstimate or use a thermometer.

Not sure about the water, as I just use tap water. The water where I live is good enough quality to not have to purchase any. (Not to mention I`d have to pretty much take out a mortgage just on water if I got into that habit, heh.)

I’m big on gravity steepers – they are big enough so theres plenty of room for your tea to move around in and give out flavor, easy to use, and easy to resteep – just add your hot water and bam! Though, this would be best if you are a 1 or 2 cup kinda tea drinker.

For water temperature, once I got a variable temp kettle like the first poster mentioned, teas changed for me – especially whites and greens! They need that low temperature! Plus, the kettle is much faster!
Those kettles go for $50 to $120ish, so it’s kinda pricey. Until then, use a thermometer, or use this as a guideline:

http://www.denstea.com/perfect_brewing.html
The best method to reach the desired temperatures is to bring the water to a boil then let it cool to the desired temperature. One way to do this without a thermometer is to pour boiling water into the teacups to be used. The temperature of boiled water cools to approximately 180°F when you pour it into a cup at room temperature. If you transfer the water to another cup, the temperature cools down by another 20°F and you have the water at 160°F. By doing this, you can have water at 180°F, 160°F or 140°F

I don’t have as much experience to offer as some of the other people here, but I’ve run into some of the same questions/challenges and can offer my solutions to them.

On the question of a gaiwan — if you are only brewing for yourself you can approximate the brew you would get with a gaiwan using a teacup/mug and a deep brew basket. David Duckler from Verdant tea always mentions this as a alternative way to make a gongfu brew with western equipment. Just get a brew basket tall enough to reach the bottom of your cup (so the leaves have plenty of room to expand in) and fill it up with ~4oz of water and pull it out after a couple seconds (or however long you’re brewing for). It’s obviously not a great way of serving a group, but if you’re going to be drinking the whole brew yourself it works great!

I’ve found that with resteeping the length of time you can leave it out for really depends on the tea. Oolongs never seem as flavorful to me if I leave them overnight and come back to them, but I just did that with a Pu’er tea last night and found it to still be brilliantly flavorful this morning! It’s really all up to taste and preference and the tea itself, but you’re always going to get more out of them the less you let them sit between brews. That doesn’t mean you have to rush though — I’ve had many a basket of oolong last me through 6 steepings over the course of an 8 hour work day with no noticeable loss in flavor.

For water temperature, if I’m really worried about the temperature because the tea is really temperamental I’ll use a thermometer, but otherwise I just kind of guesstimate. Once you measure the temperature for enough of your brews you will probably start to notice the sound the water will make as it heats to certain temperatures or how long it takes to cool to a certain temperature and you won’t need to whip out the thermometer unless it’s really important.

And most bottled water just comes from a tap somewhere, so if you have decent tap water I wouldn’t bother with it, but that’s all a matter of preference.

If you’re worried about your water you could try a water filter that attaches to your faucet. Mine has minerals in the filter to help the taste. I have definitely tasted and seen a big difference in my water since I got it, and the best part is that they’re cheaper than constantly buying bottled water. I got mine for less than $30.

1. My gaiwan was actually very inexpensive, perhaps 10 or 15 dollars with tax. If you are interested in trying it, I would recommend steeping your tea that way. It is a bit more effort but if you are keen to do it, it is nice to have that method available. There are videos available which can help you master the pouring technique, which is where I had trouble. This video from Verdant Tea really helped: http://verdanttea.com/tv/how-to-use-a-gaiwan-what-is-a-gaiwan/

2. I leave my tea leaves in the teapot (or gaiwan) for several days before I toss them. I prefer to re steep within a few hours or a day, but a couple days has yet to hurt me. I suppose your climate would have to be considered but I live in NS and have yet to have wet tea leaves go moldy, even after a week or two (that was an accident. I did not resteep after that long. :D)

3. I got a kettle that has temperature controls. Mine is a Sunbeam and cost around $40. Not too shabby.

4. Sorry, can’t help on this one. I just use my tap water as it is surprisingly good.

i use tap water, we (Netherlands) have among the highest standards for tap water in the world.

I’m with Awkward Soul on the gravity infuser. I have Adagio’s IngenuiTEA and Teavana’s version. Both are stupid simple to use.

I also have an electric kettle with variable temps. I use this one (http://www.target.com/p/oster-1-7-l-variable-temperature-electric-kettle/-/A-14024617) from Oster, simply because it was the cheapest one I could find that has a digital temp gauge. I got it on sale at my local Target store, but they have it on Amazon as well.

As a fellow New Englander, my advice on water is just to get a Britta filter for your tap. It’s the cheapest way to get clean water and seems to work just fine for my teas.

I’ll one up the $10 gaiwan… just saw some for $8 at Bird Pick today. http://www.birdpick.com/accessories.html?limit=all