Dependant on what plants you're looking to grow, it can be a difficult job getting good consistent growth.
Firstly, to get your wood to sink, you can boil it which will help it to absorb water faster so there's no need to weight it down. When boiling wood, you'll release the tannins which will stop the water being stained brown/yellow (unless you're going for that look).
Grasses come in a huge range and can be tricky to grow. I assume that you're talking about 'carpeting' grass which is notorious for being a difficult first plant to grow. You'll probably need to inject CO2 to get a good strong carpet going and you'll need to have a good watt-per-gallon ratio (more on this later) to get the light to the bottom of the tank. If you want to plant something like 'dwarf hairgrass' you'll need a good substrate (probably something like Active Substrate). Watch out for 'root compression' if you're going to use a very fine substrate with a small grass plant.
To ensure the best growth, you need to carefully balance all of the required elements if you're going to proceed with a carpeting plant. You can use this fantastic plant diagnosis chart to determine which nutrient you're missing if you're having problems but be patient and don't toss all of the different supplements in if you're not sure.
Planting on driftwood
This is a good option for your first plant (especially in a smaller tank). It's relatively low maintenance and you can usually even pick up 'pre-planted' branches at your local fish store. Plants which get their nutrients mainly through their leaves are a good bet. You'll commonly fine Java Moss, Java Fearn and Anubias (amongst many others) are frequently used. If you're going to do it yourself, just make sure that the plants are firmly attached with string. Some people also choose to use Silicon or something similar but make sure you check that whatever you use is aquarium safe before putting it anywhere near your tank.
In my experience, one of the biggest factors in good plant growth is the light you provide. For grasses which are at the bottom of the tank, you'll need a powerful light to make sure that it can reach all the way to the bottom of the tank. Here's a good guide on what lights can help your tank plants thrive but to summarise: make sure you size your light to the requirements of the plant. There's various scales but for best growth, do the calculations (don't forget the affect depth has). For the classic easy-to-grow plants, I've found that 0.5 watt per litre is perfectly adequate.
You may not want to make it too hard for yourself at the start. There are some plants that will just grow no matter what conditions they are in (within reason). Some of the easiest I've come across are:
There's plenty which grow in low light and are hardy enough to avoid the beatings of fish, poor water etc. I'd definitely start with something that will grow more easily and then graduate through the different difficulties as you gain more experience. I've bought a great looking plant, put it in my tank and watched it die despite my best efforts and it's not that enjoyable.
Plants can really make or break the look of your tank so decide how you want it to be aquascaped and then read up on the requirements for the specific plants you've picked.