What is the optimal grid size for high density commercial / residental areas? (also in order to avoid too many cross roads for traffic jams)
The largest growable building footprint in the game is 4x4, so if you want to maximize the efficiency of your grids, you'll want to make sure the greatest distance between two parallel roads is 8 tiles, otherwise you'll leave a gap in the middle that can't be filled.
As far as the perpendicular roads go, that's going to depend on your situation. Residential and Office zones produce less traffic congestion, so larger grids with fewer access roads can work just fine (Some of these zones in my cities can be as large as 20x8 or 30x8). With commercial, you may consider smaller grids, as it attracts more traffic. Also, as suggested in the comments, a "brick" pattern using three way intersections is helpful, as it eliminates a potential left turn that can back up traffic.
The main thing though is not to attach too many intersections to your large, "main artery" roads. You need to find a balance between giving the AI options on ways to get into and out of the grid and onto and off of the main roads to avoid bottlenecks, but you also don't want them stuck trying to get through three dozen traffic lights to get to the store.
Someone Did The Math (Practical Engineering: The Optimal Square Grid by me22.ca) and I have reproduced that content here - note that their numbers refer to the size of each square including roads, but I will use the sizes between the roads as this is more natural to most.
The most optimal square grid size is 10x10 with standard 2-lane roads which leads to a density of 66.666...%. This is also the cheapest in terms of road-building.
The most obvious 8x8 grid is "only" 64% efficient, which may not sound much worse, but requires 50% more road length and is therefore that much more expensive than the previous option.
For grids bounded by 4-lane roads, the optimal size is 12x12 but density drops to a meagre 50%.
You can get better density by using rectangular zones instead of grids - e.g. 8x14 is 70% density and 8x30 is 75% - but there are obvious disadvantages to this, notably in terms of traffic flow.
I use the 'Battery' Method for starting (Google YouTube Battery City Skylines). This provides a 4 x 8 grid, either side of a large one-way system.
Most importantly, turn off Traffic Lights ALWAYS. When Traffic lights went out in London, it was found that there were fewer snarl ups on the roads and traffic moved more smoothly - it's worth remembering that. The reason is that a red light will stop traffic, even when it's not impeded, whereas no traffic light will not.