The best advice I can give you is to just be consistent. 15 months is pretty young for formal discipline, but it is not too young to understand the word no or start with the 123 Magic and time outs. Buy the book 123 Magic or try watching the video (you can most likely rent a free VHS from your library). I understand that alot of people will feel your son is too young for this, but if he is a handful now--he will be twice the handful in a few years. At this point, if redirection is not working you can tell him no, give him warnings (say no a few times) and then remove him from the situation. You can put him in the playpen and give him zero attention for a minute or so when he is having a tantrum. Any attention you give to his outbursts, will just encourage him to have more outbursts. I think removing your attention is the best "punishment" at this age when they are acting out. Yelling is not going to work and you will be even more upset. Putting him in his crib or playpen for a minute and ignoring him will not hurt him. Also make sure that when he is acting good you give him lots of attention to show that is behavior you like. Basically when he is whining/tantrums ignore and when he is happy then engage him more. Of course he is just a young toddler and you need to make sure he is not crying because he is wet, hungry, sick, etc. . .but I am sure you have already addressed all his needs and are looking for suggestions on how to curb these things once all his needs have been met. He may not get this all at first and it won't happen overnite, but consistency is key.
My next suggestion is is its in your budget, can you consider putting him in some sort of setting soon with other kids? Not a large group, but a few other kids his age to let him see how a group interacts. He will still mostly be doing side by side play, but sometimes just being around other kids helps. Also you may find a great provider/teacher that has experience with "high maintenance" kids and has some tricks up he sleeve =) When I ran a daycare, I of course had some kids that were more difficult than others, and what really worked was structure and consistency. We were very on schedule with naps, meals, play time. The kids that had behavior issues did better when they knew what was coming next--surprises totally threw them off their game. Even at the age of your son, they did better with a very strict routine. The more lossey goosey, the worse the tantrums and crying. Some kids can just go with the flow--some can't. Sounds like yours is one of those that needs a set routine, so I would have one and stick to it like glue if I were you.
My last suggestion is to get a mom who has had kids of her own to babysit for you and hubby once a month at least so you can get a night out and regroup!
I hope I have helped a little. Just be consistent, get a firm routine, and start laying the groundwork for discipline techniques like time-outs now. I hope it gets easier. If it make you feel better, my dd was a "high maintenance" infant, but is an awesome preschooler! So well-behaved now, but when she was a baby there were days when I thought she would cry forever! I wish you the best.