I took a pregnancy test yesterday, and it came out positive. Is there anyway I could find out how many weeks I am without going to a doctor?

You can get a rough idea if your periods are regular. There are two ways of determining how far along you are... gestational age and fertilizational age. Gestational age goes from the first day of your last period... counting forward 40 weeks... you may not actually be pregnant the first couple of weeks or so... this is why you have a 2 week (although some now give a 3 week) window before a due date and 2 weeks after a due date as a normal time to deliver. Due date would roughly be counting back 3 months from the first day of your last period and then subtracted 7 days. Fertilizational age counts from when you actually conceived... this one gets a bit more trickly unless you know when you likely ovulated.
Given all of that, you really should go to the doctor to start on your prenatal care... they can do an internal exam to get a rough idea from what your cervix feels like, and later a scan will give an indication from measurements that are taken. OR visit a health clinic, or planned parenthood if you need discussion about your options.