It's going to be different for everyone, but... I've been on HRT for 3 months now, still a pretty low dose I think, but here's what I've noticed.
I noticed a change in my state of mind and happiness levels immediately, like, within a few hours, and then it evened out over the next two months, to where I just feel "normal" now, happiness-wise.
You will probably get emotional and have mood swings and hot/cold flashes in the first few days, which can last for weeks before evening out as well. Then when your dosage changes, prepare for them to happen all over again. And you may still get the occasional "PMS" spike just because.
I started using a tape measure to track changes to my body shape, and have seen an inch or two here and there move/change over the months. I only check once a month or so though.
I "feel more girly" now with almost every passing day. At first it was kind of a placebo thing I think, sort of. I was trying harder, so I perceived more changes to my attitude and mannerisms, etc.. Over the past month though, it's become automatic. Hanging around other girls and wearing girl clothes more frequently has really helped with this, it's more natural now. I don't know how else to explain this. I sometimes worry that I'm "overacting" though and coming across as a gay man in my day-to-day work life.

There is a notable difference between a guy acting feminine/gay and a the way a girl acts, I just don't know how to describe it.
I've had people tell me my voice is naturally becoming more feminine in cadence, breathiness and tone as well. Pitch slightly maybe, but less noticeably so. I fortunately didn't have a very deep voice to start with. I also pickup accents pretty easily, so this may just be the way my brain is wired.
A friend I haven't seen for a month now commented on all of these things just the other day, she's impressed with the new girly vibe I give off, haha.
Facial changes, maybe... It's hard to tell. That area is going to be much more subtle. My skin is improving, softer, smoother, less oily, and my pores are getting smaller. Everyone says I look fantastic!

I think my eyes looks bigger too, maybe just because I'm happier to greet the world with my new self though!
OK, and so the big thing on every newbie's mind, boobs. My nips have hurt for the last 3 months. That started in the first two weeks. It gets worse and then better, and then worse again. Sometimes it's aching like a sore muscle, sometimes it's itching, sometimes it's a warm tingling, sometimes it feels like needles. These sensations come and go almost daily. You get used to it. Buy a sports bra (they make them with a zip-up front and they are great when you have bigger shoulders like most of us), and wear it whenever you are carrying boxes or anything that requires you resting something on your chest. Sometimes I can't even sleep on my stomach at night anymore because of the pain. And prepare for everything in your environment to crash into them at some point, they're like magnets for stuff to bump into them now.
I was lying in bed a couple nights ago, on my stomach and propped up on my elbows, texting some friends, when I looked down and was like, "OMG, BEWBS!!!" They're tiny, but there is definitely dangly breast material there where there wasn't before. It seems to have mostly developed over the last month. I can pull together some decent cleavage with a sticky-bra now, hehe.
So there you have it, my 3-month HRT report. Remember, everyone will respond slightly different, but the gist of it is that you will see minor changes in almost every area probably. There won't be any noticeable changes except in your mood and possibly personality, but that's because you're happier! Everything happens slow and subtly.