In terms of the process of beginning an actual excavation, the first step is to apply for a permit. All archaeology has to be regulated by the government, because keeping records is incredibly important (you can't dig the same dirt twice). In order to get the permit, you have to outline all your excavation strategies, the questions you want to answer, and account for how you will conserve the artifacts you excavate. It's a way of keeping archaeologists accountable, so that you don't end up with hundreds of unpublished excavations languishing in various people's personal archives. Publication is key, but most archaeologists prefer the fun digging part to the boring writing part, so getting them to publish is sometimes like pulling teeth.

After this, you organize your resources, the labor force and the equipment you will need. Occasionally, if there is significant overburden (natural accumulation above archaeologically relevant material) it is sometimes necessary to make arrangements for larger equipment (bulldozers, JCBs, etc) to remove this dirt so you don't waste precious time excavating dirt without remains in it.

Once you have your permit, you have to set up your orientation and mapping system. Basically, you need to establish a place from which to reckon all the coordinates of all the things you are digging. This could be in UTM coordinates if you are doing a larger-scale survey project, or you can establish a local grid with a known tie-in to global coordinate systems. Then you decide where you want to start digging, mark out your areas and map them in, and then start excavating. Getting the spatial recording set up is very important, you need to be able to map anything in the moment you find it. Again, because you can't dig it twice, you have to get it right the first time. More records is better than less records. Record everything.

When you do start digging, the exact strategy will depend on you geographical location and the nature of the site you are excavating. In the US, often the first stage of excavating is probes and test pits to figure out the location and extent of the site, followed by slightly larger probes to determine the best place to do a full scale excavation. In the Near East, this sort of extensive probing is not as necessary because most sites are very obvious.

As for which grid system you use, that is highly dependent on the nature of the site and your excavation strategy. The main strength of the Wheeler-Kenyon style box grid is the consistent baulks between excavation areas, keeping a consistent reference to vertical stratigraphy which is left behind. Open area excavations have to rely on the records you keep to refer back to previous strata, while the box grid method has those previous strata available for visual reference. It is important in both methods to excavate in a manner that preserves occasional "sections", either in the middle of the excavation or on the baulks, so you can see what you are about to excavate and can make more informed decisions. For example, if I am digging a pit, or a room, I will dig half of it first, and check the section to see if I have missed any changed (which you should see in the preserved half), and if I have, I can correct that mistake on the other half of the unit I am excavating. So keeping vertical sections available helps with visualizing stratigraphy and preserving the vertical component.

However, on larger sites, when the excavation gets particularly deep, these tall sections can become dangerous, so they must be partially knocked down. It always seems to happen that the most interesting artifacts and most important finds are discovered in these baulks, and it's sort of a cliche among archaeologists that if something important is found, "it's in the baulk".

Part of the difference between the two methods is how one conceives of the material you are excavating. At the excavation in which I work, each separate depositional feature is given a different unit number, so a fill on top of a floor is given a number (this unit might be a meter thick), while the floor and subfloor are given different numbers. All the artifacts from this depositional unit are numbered and tagged as being from that unit. Walls and foundation trenches are given different numbers. Pits are given two numbers, the cut and the fill. Because each depositional unit is distinguishable from the others and can be drawn the same day it is being excavated, there is less need for frequent vertical stratigraphic reference points, so we don't have 4m x 4m box grids everywhere. We do maintain baulks every 10m, and they are recorded. Sometimes, excavations where there are numerous walls which divide up the area use those as division points, and are less concerned about the vertical aspect. This is a methodological difference. I personally think that the vertical aspect is important, and efforts should be made to record the vertical aspect of an excavation, and the resolution of those points (4x4 m box, or 10x10m box) will depend on the sort of site you are excavating, whether you are dealing with small-scale campsites with lots of organic features, or larger-scale urban settlements with massive stone walls.

In the end, it's all about the way you can get the most useful information in the shortest amount of time.

LIDAR is a surveying method that uses a laser beam as "radar" to scan the surrounding landscape and create a three-dimensional model. Basically it takes thousands of measurements all over whatever it is you are measuring (such as a landscape) and interpolates between those points to create a surface which models your survey subject in three dimensions. It can be set up as a stationary tripod to scan an area near it, or it can be operated from an airplane in order to map a larger landscape area. It uses EDM (electronic distance measuring) by comparing the time delay that a laser reflects as it comes back. It's very precise, and is the same basic technology that all modern surveying Total Stations use. An internal computer triangulates the measured position based on the internally stored location (from when you set up the machine, or stored via GPS information) and the angle/orientation that the laser is pointing and the distance that the laser measures.

I hope this answers your questions, if you have more, please ask away and I will try to answer them as best I can.

Hi! There are a few archaeologists here, but fyi, you might get more input from /r/Archaeology or /r/AskAnthropology