Hey damnable.,

Climbing out of the saddle is one of the most physically tiring parts of climbing, so don't worry if it takes some time to build up your strength.

From what you mentioned about not being able to stay out of the saddle very long, it sounds like you may need to strengthen your core muscles (abs and lower back) a bit, as these are what support your weight when out of the saddle. Doing some sit-ups, back extensions or deadlifts, and climbing out of the saddle more will all help with this.

You're dead on with your comment about there not being enough resistance in the pedal stroke. If I'm comfortably pedaling seated on a hill, I usually upshift 2-3 times as I get out of the saddle. Your cadence will naturally drop by 20-30 rpm when you do this, but I think you'll find it much more comfortable than spinning a small gear.

Most importantly, just keep practicing. Every time you do a climb, get up out of the saddle for as long as possible, then once you're spent, sit down for a minute or two, then repeat. After a month or two of doing this (depending on how much time you spend climbing hills), you'll find that it gets a lot easier quite quickly

With regard to weaving, how fast are you going when this starts to happen? If you're going really slowly (maybe 5-6 mph), then this could be the cause. Otherwise, try and lean forward a little bit. A good rule of thumb is to have your head directly above the intersection of the stem and handlebars, and try and get a rhythm going, with the bike rocking gently back and forth through 9-12 inches of arc. Again, this will be easier if you're pushing a bigger gear at lower cadences.

Most of all, keep doing hills, they're what make cycling such a beautiful sport

EDIT: sorry if you know some or most of this already, but there are so many different levels of knowledge, best to have too much information than not enough