After I am settled in on my weight for XXX, how do I know it is time to up the weight?
If you're a novice I'd recommend doing Starting Strength, which is what this thread is about. If you do that, then it doesn't get much simpler. You do 3 sets of 5 reps and up the weight 3 times per week.
Someone suggested when I can do it 15 times, slowly, without grunting and groaning.
This is absolutely ridiculously horrible advice. You go at the speed necessary to complete the lift. And if you're not grunting or groaning then the weight is too light. Lifting heavy takes a strain. Noise is a product of that strain.
Then the next question is, how often should I expect to up the weight?
The next question you need to ask yourself is "what program am I going to use?" It sounds like you're just trying to play it by ear and go by feel, which is a recipe for disaster and failure. And it sounds like you're a novice at this.
It's real simple. Buy a reputable program by a known and reputable strength coach, and follow it exactly. I'd either get Mark Rippetoe's Starting Strength:Basic Barbell Training or Jim Wendler's 5/3/1.
If you don't know how to do each lift very well with excellent form, by SS:BBT. The 2nd volume being the most popular and widespread and the 3rd being the newest. Either of those will work. They will teach you exactly how to lift each lift properly and provide you with a great program that is easy to follow. It is designed to get a novice, even people who can only lift the bar, as strong as safely possible in the shortest amount of time. It can be tough since it is 2-3 main lifts per day, with multiple sets of 5, 3 days per week every week. But it works if you follow it.
5/3/1 is a great program but the book isn't truly aimed at novices. It barely touches on program technique and form and goes straight into programming. Wendler's book is aimed at those who know how to do each lift already. It can work just fine for novices and is less grueling and demanding than SSBBT, and more flexible with the schedule, allowing for 1/2/3/4 day per week programs. It also works well, but you've got to at least know the proper form to start.
I'd recommend both books. Even if you don't use the SSBBT program, it's an invaluable tool that can be looked back upon anytime you run into an issue. And Wendler's book provides a program that can be used at any level, and restarted at any point. It's literally capable of taking you from novice to elite.