The clinic can be expanded with more ambulances and more wings, remember to consider the extra space a wing will take when plopping, these are shown by yellow or red markers around the building before you plop it. Red indicates no room for expansion in that direction, yellow means yes you can expand your building in thtat direction.
Got a lot of money, and know what you're doing, opt for the hospital straight away!
Upgrading your Health Service
Once you've outgrown yor clinic and added all of the upgrages (more ambulances and waiting rooms), it will be time to plop a second clinic or a hospital. My choice would be hospital as I find that Sims tend to get sick quite often. There are also extensions that come with the hospital to promote health, decrease germs and provide 'Wellness' services to it's inhabitants.
Assuming you have just started to play the game and have no other cities in the region, the only option available to you is to plop a clinic. Select the health icon and you'll see the clinic building, select that and hover around to try to find the most suitable location for placement. The green shading over the roads will indicate the level of coverage for your new clinic. Eventually you'll need to add a couple of these (or a hospital), so when plopping you should consider where you're second clinc or hospital will end up in relation to the first. I mean don't just stick this smack bang in the middle of the map as your second clinc may overlap in coverage. Maybe chose a residental area towards the top left of center and for your second later int he game the area towards the bottom right of center. Laying a clinic will have a positive effect on your residential population, they will be happier and as with fire and police serivces, it will add towards the calculation which determines density and wealth.