From the way you mentioned your "problem" I get the impression you are not a local girl which is such a pity, but that was me and yes, it is very hard and very worrying trying to fit in with a crowd of girls that know everyone, except you!!
You need to find out when and where all the different toddler groups are in your area. Then try out each one that are easily accessible to you. Ideally you want to make new friends with girls who are at toddlers stage just like you. Not every Wee one is walking by their first birthday, so don't think...ever, that your Wee one is slow.
Then over time, but you do have to be patient, you will eventually click with someone and suddenly a new world will open up for you both. You are not alone at feeling desperately sad. If every mum was truthful, she would admit there are times when she feels just as low and that things aren't going the way she dreamt they would.
Goodness, you are young with a young baby, with the most exciting time in your life just starting. Be kind to yourself, you are doing the hardest job in the world and you don't get paid for it!! You need friends for your toddler, but friends for you also, so be brave, someone has too do all the initial first moves.....say all the hi's, hello's, how are you, to all the girls that you instantly like the look of because that is your "chemical instincts" trying to match you with someone that really is very much like you.
Remember YOU are a nice person, and it is others who are probably much shyer than you, so you make their day by saying that first hello with the biggest smile you can make......and enjoy your baby!!!