Regular fluorescent tubes in a shop light fixture are just fine, BUT they have to be hung so that they will be just 2-3 inches above the plants and can be raised as the plants grow. The fixtures usually come with hanging chains. If you hang them from hooks, then you can just put different links of the chains on the hooks as needed. Leave them on 16 hrs a day.
Maybe you don't absolutely, absolutely need a heat mat (or some source of heat), but since I've been using two of them for years, I wouldn't try it without any more (depending a little on how warm the area is where your lights etc will be). Not only will your seeds germinate three times as fast, but you will have a better success rate. While the seeds are sitting in cold moist soil not germinating, they tend to rot out.
But you don't need a fancy one made for plants. What I use is ordinary heating pads sold at a pharmacy for people to use. Just be sure it doesn't have automatic timed cutoff.
Congratulations on being the first to write in about seed starting!
I usually plant the first seeds under lights some time in January and I'm an early bird.
Here's a picture of my seed starting operation in my basement (the picture is from a couple years ago, there's nothing under the lights right now, and that area is about to become my Christmas gift basket factory). I start somewhere between 500 and 1000 plants from seed each year.
On the bottom right, the little blue lines are the edges of heating pads. Once the seeds are well sprouted and have true leaves, they can come off the heat pads.