A simple model of an object's properties can be expressed with first-order predicate logic. This model assumes that we have a set of logical objects, and certain predicates these objects satisfy. If b is an object and P_ is a one-place predicate then if Pb is true, then b has the property expressed by predicate P_.
If we restrict ourselves to one-place predicates, we have what we could call non-relational properties. If we look at predicates with two or more arguments we get relational properties. For example, b is to the left of c, is not a property of either b or c taken in isolation but a property of the two in an ordered relation bLc.
However, this logic-based account of properties may be metaphysically implausible. For example, if Pb is true for any predicate, then Pb ∨ Q is also true for any proposition Q (here "∨" is the symbol for logical "or").
We can define a new predicate P' as:
P'_ if and only if P_ ∨ Q
It is easy to prove that P_ implies P'_ since P'_ is always true if P_ is true.
Yet, it might seem implausible metaphysically to say, for example, that an "an apple being red" implies that the apple also has the property of "being red or pigs fly". Thus, some philosophers might require a more substantive (evidence-based) argument for asserting that a given predicate P_ expresses a real property. We can still say that all properties can be expressed as logical predicates, but not all logical predicates express real properties. It is then the job of the metaphysician to determine which predicates correspond to real properties on a case by case basis.
The other part of the question is about changing properties. If we look at an object as existing in 3-spatial dimensions at a moment in time, it appears to have certain properties which we can call synchronic. If we consider something that has a property which holds across across time, we can call this a diachronic property. For example, the property of turbulent flow in water is a diachronic property. Now, if we are looking at a synchronic property changing between two times, in the background there needs to be a diachronic property that describes the object's capacity to change while maintaining its metaphysical identity across time. In general, an object b's capacity to change certain properties while remaining b is a diachronic property of b. Historically, philosophers have called the properties that can change without causing b to cease being b, the accidental properties of b.
Let's consider your example of Mike becoming the chef at a restaurant. We generally assume that the capacity to maintain personal identity across time is a diachronic property of persons (although this could be challenged). For example, when I wake this morning I'm slightly different than I was before I went to sleep, but I'm not a distinct person. Certain kinds of change can cause a loss of personal identity, for example, if a person dies their body ceases to be the person they were although the body continues to share some of the properties that were previously possessed by the person. However, in Mike's case becoming a chef is not a sufficiently radical change for Mike to cease being himself. In this case, Mike simply gains the additional property of being a chef.
