So more back-to-basics here: Skip this if I'm preaching to the choir.
There is "checking" for looseness for movement, then there is "mechanic's feel," then there is critical feeling and measuring.
One needs to practice the technique of moving something carefully, with barely any large-muscle engagement. If one holds on tight and tries to move something with all their might, the nerve endings will not be sensitive to small motions. This is a skill worth developing, but it is in the realm of techniques that take a while to learn, but stay with one once learned. An an example of an equivalent skill and technique is hand-sharpening things: It is rarely learned in one step, but is worth learning over time.
If one has the "feel" for fits and motions, it goes a long way toward getting to the root of problems like the one(s) under discussion.
There is also the need to know how the machine under test is constructed and to think through the possible places to look for play and consider what may be out-of-adjustment, or needing modification.
That said, there may also be the need to move on to measurement. In measurement, there is usually only one dimension of movement to check at a time, with careful thought as to what is actually being measured.
So, on my mini-lathe, I carefullly try to move and feel for motion of the compound relative to the cross-slide, then tighten the screws slightly and do the same check again. Rinse and repeat. Then I move on to the cross-slide and repeat everything again. then I move on to the Saddle and do it again. Then, when I thiink I have the problem isolated I apply the indicators, etc and check again.
Now, if I have isolated the problem, or the worst of the problems (which is what one does with an inexpensive machine), I take things apart and start measuring parts the best I can with the tools I have. The how-to-do-it of these measurements are explained in Connelly and other books.
There are measurements of a part with scale, square, micrometer, etc.
There are checks for straightness and squareness against standards like parallels, squares, and surface plates.
There are checks for fit with guages, shims, and dyes.
After all that, there may be the chance to do a modification, but tread softly and keep checking.
Of course, there are also the issues relating to mounting the machine, leveling it, and using it within its limitations.
What one finds with an inexpensive machine is that there is always room for improvement and one generally works through tests and problems in a spiral that gets progressively closer to having a well-adjusted and precise machine. What one also finds is that one will eventually get to the place where not much more can be done and "better" is probably a better machine: I still have my Mini-Lathe, but I can't turn it into a Monarch or a Hardinge. I just have to work around the problems with what I have.
Again, apologies for droning on with this.
--ShopShoe