Black thumb? Hm. I think the notion of green or black thumb is wrong. Anyone can grow plants, everyone will kill them too.
I could write a book on the subject. I'll try not to here

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The first important thing for being successful with plants is soil.
Soil does a few things:
1. It provides a loose mass for the plant to root in.
2. Soil holds water and nutrients absorbed by the roots and 'pumped' into the leaves where with the addition of sunlight converts it into building materials for the plant.
So having a good, healthy rich soil is the first key to healthy, vibrant plants.
There are perhaps hundreds of potting soils on the market. These are great for one or two potted plants where you will feed and water on a regular basis.
However potting soil is mixed with filler to make it light weight, water retaining - then it is kiln cooked to kill all bacteria and possible diseases.
I have through experimentation come up with a good soil mixture:
Rough Construction Sand.
Composted organic material.
Peat Moss.
I'll give you ratios here:
2 Sand
3 Compost
2 Peat-moss
I mix these together well.
The sand provides a good weight to keep the pot upright, it also provides drainage. Sand is basically dead weight, it does not hold water, and it does not hold nutrients.
The compost (which you can see I use a lot of) provides good food and will do so for several years even if I do not feed the plant.
Peat moss is super dry when you get it, however it absorbs and holds water for a long period of time, it will decompose adding more food to the soil.
How you prep the pot for plants is instrumental for healthy plants as well.
I like to cover the hole at the bottom with a stone or something that will hold in soil. Then I mix 1 part soil with 3 parts pea gravel or similar and cover the bottom of the pot with an inch of this material.
My guide for deciding how deep that first layer will be is the water tray I will be putting the pot in. The desired depth of that first layer is deep enough to where when the tray is full of water the first layer of soil and gravel is at water level.
Then I plant the plant, adding soil as fill leaving about an inch at the top of the pot or more.
The first time I water I set the pot in a bucket and fill the bucket with water. I let the plant sit in that about an hour. Then I pull it out and let it drain.
I put it in it's tray and fill the tray with water.
From then on out I water the plant by filling the tray.
Most people will pour water from the top - all this will do is wash the nutrients out of the soil down into the tray. If you fill the tray, the water will slowly wick into the soil, keeping the soil moist. I water when the tray runs dry.
The only time I water from the top is when I am using a compost tea or a liquid plant food.
As for an automatic watering system, Drip irrigation is available.
You can get the tubing, fittings etc at any home store - or you can buy whole kits.
http://www.jessstryker.com/ May be very helpful and a good place to start. However you will find that there are many books on the subject as well. I found that there are a lot of sites which sell these systems as well. These can be made into automatic timer systems rather easily.
As for what kind of plants:
http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/webdoc1375Appears to have a good list of house plants.
Cheers
David