I generally use grass cuttings and vegetable waste. When material is added should it be gently compressed or just left as it had been added
You should definitely not compress what you add. This is because you want oxygen to get into the composting mix. You will find that over time the compost pile will compress itself as the material breaks down, and it should in fact be turned in some fashion to restore the pile's access to oxygen. A compost pile without oxygenation turns to "anaerobic" decomposition, which can emit unpleasant odors, and breaks down the material slower. If you've ever just piled grass clippings alone and left it for a week or two and then dug into it, you'll see that it forms slimy slabs of smelly material. This is anaerobic decomposition.
I would use a search engine (like google) to query "composting for beginners" and read up. It's not complicated, but there are important details necessary for a successful compost effort.
Compost needs 3 main things to work well:
- The proper mix of materials that are carbon-heavy and nitrogen-heavy. The pile needs 1 part nitrogen material to 2 parts carbon material. Grass clippings and food waste are both very nitrogen heavy, so you will need to find a source of carbon material to balance out your pile. I suggest dried leafs or wood chips. The materials should be well mixed.
- Moisture. Take a handful of the pile and squeeze it in your hand. If your hand is dry your pile is too dry. If your hand drips water out as you squeeze the pile is too wet. Adjust appropriately and be sure to mix.
- Oxygen. Most home composting is done as an aerobic or oxygenated process. You need air mixed in. Grass, in particular, is well known for causing clumping and pockets of a pile that do not have enough air. Your nose will let you know because if the smell is bad then it's likely not enough air mixed in. Some small twigs or stems of hardy plants (e.g. sunflower stalks) can be helpful in creating an environment where air flows freely. Add in a mix of material to help the airflow and turn the pile to get it oxygenated.
Do these things and your compost will be warm and quickly turn from the raw ingredients to a healthy, garden enriching compost.