What you should do is highly dependent on what you are looking at and why you want to make those changes. It also depends on which method you want to use
There are quite a few different ways of doing this for mammalian cells. One commonly used way is to have the gene expressed off a plasmid in a cell line that lacks expression of the protein of interest. This means you have an easily manipulable plasmid which you can make and confirm sequence changes easily and quickly. Plasmids are also easy to use and can get results quickly. However, these are usually non-permanent changes, unless you have a selection mechanism like geneticin and maintain this on your cells. If you don't maintain the selection the cells tend to lose the plasmid. The downside is that you don't tend to get every cell transfected and the percent of cells transfected drop with plasmid size and/or method and can be variable between different transfections and different plasmid preparations. These methods are known as transient transfection and stable transfection respectively. For many sequential changes, looking at intermediates, this is probably the method I would use, but CRISPR/Cas (see below) is also promising.
For targeting of the genome of the cells you are looking at, conventional gene editing techniques by recombination between plasmid and genome are slow and best suited to smaller inserts and low numbers. They tend to have lower success rates and require things like antibiotic selection to ensure that they have happened. This includes things like Cre-Lox recombination. These tend to be quite slow relative to transient transfections and require you to have robust sequence checking methods in place (often done by PCR product size and/or sequencing), but are stable and don't require continued selection once stable. They can be done at a single-cell level and can be site-specific, depending on the method used.
Viral transduction is similar to both transfection and recombination methods. You usually work with relatively easily manipulated virus genomes expressed off plasmids to make the viruses, which are then used to do the recombination/expression. It does tend to have quite a bit of finicky set-up, but is robust and stable once it is working. Lentivirus and Adenoviral vectors are two common methods of this, but there are others.
CRISPR/Cas I don't know a lot about, but I know it is entirely possible to use this method to rapidly make site specific changes in genomes. I think this is the most promising technique for many genetic manipulations in the cell. This is very quick, but requires quite a bit of optimization.
In general, making lots of changes at once is entirely possible, though, as you will have realized from the above, this depends on the mechanism used to make the changes. Just be aware, that if you are manipulating the genome of a host, dramatic changes (i.e. many) might drastically affect the cells in some fundamental manner so that you don't get any cells survive. This is where intermediate steps might be useful, but is something you can choose or not at this point.