I'll give a very qualitative answer / overview.
The classification 'first-order phase transition vs. second-order phase transition' is an old one, now replaced by the classification 'first-order phase transition vs. continuous phase transition'. The difference is that the latter includes divergences in 2nd derivatives of $F$ and above - so to answer your question, yes there are other orders of phase transitions, in general.
Note that there are phase transitions that do not fall into the above framework - for example, there are quantum phase transitions, where the source of the phase transitions is not thermal fluctuations but rather quantum fluctuations. And then there are topological phase transitions such as the Kosterlitz–Thouless transition in the XY model.
The framework to understanding the thermal phase transitions is statistical field theory. A very important starting point is Ginzburg theory, and then you upgrade it to Landau-Ginzburg theory. In a nutshell, phases are distinguished by the symmetries they possess. For example, the liquid phase of water is rotationally symmetric and translationally symmetric, but the solid phase (ice) breaks that rotational symmetry because now it only has discrete translational symmetry. So there must be some phase transition between these two phases. Liquid and gas possess the same symmetry and so actually can be identified as the same phase, as evidenced by being able to go from liquid to gas by going around the critical point instead of through the liquid-gas boundary in the phase diagram. LG theory involves writing a statistical field theory of the system respecting symmetries of the system, and then studying how the solution to the field equations respects the symmetry or not against the temperature.
Now we don't really deal with first order phase transitions as much as continuous phase transitions. I can give a few reasons:
First-order phase transitions aren't very interesting. You can model them by Landau-Ginzburg theory in the mean field approach by adding appropriate terms in the effective action (like $m^3$, $m^4, m^5, m^6$, $m$ being the order parameter [yes, note that odd terms are allowed - they explicitly break the symmetry. Although for reasons of positive-definiteness the largest power must be even.]).
First-order phase transitions depend on the microscopic details of the system, so we don't learn much information about such a PT from analyzing one system.
Or perhaps, we just don't know how to really deal with first-order phase transitions very well.
Continuous phase transitions have a diverging correlation length (first order ones typically do not). This implies a few very important things:
a) Microscopic details are washed out because of the diverging correlation length. So we expect continuous phase transitions to be classified into universality classes. By that I mean that near such a critical point, thermodynamic properties diverge with some critical exponents with the order parameter, and these set of critical exponents fall into classes that can be used to classify different PTs. Refer to Peskin and Schroeder pg 450 - we see that the critical point in a binary liquid system has the same set of exponents as that of the $\beta$-brass critical point! And the critical point in the EuO system is the same as the critical point in the Ni system. Interesting, no?
b) We can use established techniques such as renormalization to extract information of the critical exponents of the critical points. Try this paper by Kadanoff.
Ok, so as I said this is a very qualitative answer, but I hope it points you in some (hopefully right) direction.
Best Answer
The short rough answer is no. The transition between gaseous state and plasma is continuous and gradual. Phase transition typically happens at constant temperature for given pressure, which doesn't happen for plasma. Have a look here.
Some references classify the transition from gaseous state to plasma as a special type of phase transition called second order phase transition.The difference between the second order and first order (standard well known phase transition) is that second order is gradual while first order is sudden. Have a look here.
So if you are referring to standard definition of phase transition, the answer is no.
Hopefully that helped