Excuse the long reply, but I'm currently consolidating lots of information to post under different titles but haven't finished yet.
I assume you've been diagnosed by a lap? Which hospital did you go to and who did the surgery? You can then try to establish whether the gynaecologist has any specialty in endometriosis. This used to be very easy by looking on the Dr Foster website as they have a registry of consultants with their field of work, but it's down at the moment.
Second, I suggest you obtain a copy of the operative report and discharge letter from the hospital which will have been sent to your GP and scanned to your medical record. You are entitled to copies of anything on there as obviously it's about you and your health. This will then help to establish what stage you are. There will be a small charge to cover the paper and ink but not a lot. I suggest you do this in writing. You are actually very lucky to have been openly given the choice by your GP as to what you want. This is your right under NHS Choices and you can go anywhere you want to the specialist of your choice, but sadly many have their GP decide which is wrong.
Basically you have been given the choice of medical management or surgery. Medical management is only considered effective in stage 1 but should be considered carefully. The medications aim to put the body either in pseudo pregnancy or psuedo menopause. Those that do the former are the combined birth control pill and the mirena coil and they alter the balance of the sex hormones to manipulate the menstrual cycle. Those that put you in pseudo pregnancy are generally medicines called GnRH agonists that act directly on the brain to stop the pituitary gland sending messages to your ovaries to stop oestrogen production. All medical management is a short term remedy to try and reduce pain, but the pill and coil can be taken the longest. The 'menopause medicine' can only be taken for 6 months as it can thin your bones and increase risk of osteporosis later on and have other (sometimes permanent) long term side effects. Once medicines stop the symptoms will gradually return as they can only halt the endometriosis for a while not stop it. Medical treatments are also reported as reducing fertility in some women.
On the other hand, surgery by an experienced specialist in endometriosis can remove all endo and theoretically resolve it. The earlier stage you have surgery the better you chance of it never returning. Unfortunately this won't 'cure' period problems as they are caused by altered hormonal activity in women with endo and not the endo itself.
There choice is yours. But my view is that short term medical treatment will only give temporary relief with worse endometriosis developing further down the line at the risk of side effects. The sooner it is addressed with surgery the least invasive that surgery will be and the better the long term outcome, in terms of pain and fertility.
On the fertility front you can take some charge of this if you want to at home. Endo is actually quite rare on the uterus, ovaries and Fallopian tubes. Infertility is much more likely caused by a short luteal phase of the menstrual cycle. This is the second half between ovulation and the period - that is to say the time during which a fertilised egg would implant and become viable as a pregnancy. Although all women might vary slightly, the accepted minimum length of this luteal phase for a pregnancy to result is 10 days otherwise the egg is unlikely to implant. In a woman with a 'normal' 28 day cycle then the follicular phase (the first half when the follicle develops) and the luteal phase will both be around 14 days as nature intended. If a woman has a 24 day cycle and ovulates at say day 14 then she only has the critical 10 day luteal phase and should try getting pregnant asap. Women with endo are often found to have shorter cycles and thus shorter luteal phases which is the main factor causing infertility as women with endo get older.
If your cycle is 28 days or not much less and your Fallopian tubes are not affected (or you have at least one viable one) then signs for pregnancy will look promising. If it is closer to 24 days then you can keep a diary of the length of your cycles (which I suggest to all end suffers), marking day one as the first day of bleeding, and do this from now on. I don't know if you have a little ovulation pain mid cycle (I did) but if you buy an ovulation tester and do this around mid cycle, perhaps from day 12, you can establish what day you ovulate on within a day or so. This will usually be fairly constant. Get a test that identifies peak fertility which will be when you have a surge in LH that triggers ovulation. Take ovulation as the day after and deduct this day from your total cycle length and you will have a good idea of the length of your luteal phase. If you have ovulation pain then you will have even better. If what is left is over 10 days then infertility is less likely to result from this cause. Cycle lengths do vary in some women but if there is an overall shortening trend then this will be a warning sign of increased risk of infertility.
I will be more than happy to help further if you get the lap report.