I have one of those babies, it's got to 6 months and there is some small improvements then he has started teething....
It is hell and I sympathize, I am a grumpy zombie and long for just a block of sleep, luckily my do is fantastic and does the school run so I can get a couple of hours of solid sleep and my body has adjusted to a point :/
Sleep training may not be the answer as he is very little still the big thing that helped is getting rid of the dummy to sleep, my ds can now sometimes go 2/3 hours at a time. Don't feed to sleep, unlatch when they are very sleepy but awake sit them up to rouse them a little then get them in their bed.
The hardest part is knowing when to remove the milk so he will settle but not too late that he is asleep, that will take some practice.
Also decide on what bedtime will be, with my ds everything that happens after 6:30 is nighttime so is done in the dark and no chatting.
Take what I'm saying with a pinch of salt though, ss is starting to improve but is not one of those babies that sleep through, his normal schedule is:
6:30pm bed
8pm brief waking
9:30 brief waking
10:30 wakes up needing milk, nappy changed so he is awake enough for it
1am awake try and settle him without milk
2:30 awake he normally takes a short feed
4am brief waking
5:30 brief waking
6:30 up for the day
During the day he will have 2/3 naps depending on what we are doing but never later than 4:30 or he doesn't settle well. The brief wakings providing he isn't upset I leave him to settle on his own but any sign of distress I go to him stroke his back and say it's ok it's sleep time. Sometimes he is ok with that other times it's pick up hug put down awake.
I found the pick up put down method work ok but it is tiring the first few nights.