Quite literally just find an access point on the Madison and start to fish. The statevaccess points are well marked and easy to find. Also, any of the shops in Ennis (last time I was there there were 3 of them in that little town of 700 people) can provide you with info on were the river access points are located. The closest access to Ennis is right at the east end of town at the bridge. the river is this area starts to braid a bit because it is just above Ennis Lake (a made made power project) and it gets quite a few folks fishing it because it is so close to town; but it can provide some very good fishing if you walk up or downstream 1/2 mile.
Also, the river below Ennis Lake (ask the folks in Ennis how to reach it, although if you get to Ennis by way of Bozeman you will cross the Madison below the lake) can be fantastic fishing in late August through November. There is a county road running along this portion of the river that provides good access to the river. Also, most of the ranches gladly give permission to tresspass and fish if you stop and ask.
Along with the great fishing on the lower river below the lake, there is a wonderful spring creek on public land in this area too, meaning there are no
"rod fees", that is never crowded. It is not easy fishing; but is has a huge population of very nice browns and rainbows that are suckers for Tricos in the morning and caddis in the evening. I will not tell you the name of this little gem; but I will tell you there is a public parking lot and access to the river and this spring creek is found on the county road (which is gravel). I lived 50 miles from this area for 4 years and fished this spring creek and the lower Madison many times.
Enjoy your trip to the trout fishing heaven found in Montana.