What is the simplest system that has both, discontinous and continous phase transitions? I am looking the simplest system that has both discontinous phase transition and a continous phase transition between the same phases (you can change one parameter). 
discontinous transition: first order transition where (some) measurable has a discontinuity/non-analyticity as a function of (some) order parameter
continous transition: measurable is continous/analytical for all values of order parameter.
simplest: Requires the smallest number of mathematical symbols and is still defined exactly c.f. ising model.
EDIT: Removed water as an example. Above critical point, water has no phase transition as Alexei pointed out. So, all the transitions of water are first order...
 A: I think you are wrong with water. Above the critical point there is no transition in water at all. This is also true for any other isostructural transition: as soon as there is no symmetry difference between the phases, in the continuous case you cannot say, if the transition has already taken place.
A correct example should be probably, KHP (potassium dihydrogen phosphate, see here: http://en.wikipedia.org/wiki/Monopotassium_phosphate or KDP (Potassium deuterium phosphate). You may look into the books of Tonkov:
1.  Tonkov, E. Y. High pressure phase transformations (Gordon and Breach, Philadelphia u.a., 1996).
2.  Tonkov, E. Y. High Pressure Phase Transformations: A Handbook: (Gordon and Breach SA, 1992).
and may be also in this one: Tonkov, E. Y. & Ponyatovsky, E. G. Phase Transformations of Elements Under High Pressure (Advances in Metallic Alloys) (Crc Press Inc, 2004).
You will find there lots of examples of various transitions, and I have also seen there examples of transitions of the second order that come into the first order through a so-called, tri-critical point. It should be the example you are after. To my knowledge, there are, however, only few such examples in structural phase transitions. There should be some more examples among magnetics, but here I am not a specialist. 
A: First, make sure to read up on definitions to clarify what you are looking for - classification of phase transitions isn't 100% science, and has a little bit of fussiness to it.  Wikipedia's page isn't terrible.
Second, I can't tell you whether it is the simplest or not, but as I understand your question, the Ising model itself satisfies your conditions, as long as you include a magnetic field (and are in dimension d=2 or higher, so that there is a phase transition!).  
At zero magnetic field, if we decrease the temperature of the Ising model from $T > T_c$ to $T < T_c$, the magnetization M(T) increases continuously, i.e. we can get an arbitrarily small M by taking T arbitrarily close to Tc.  Now, suppose the model is below Tc and we have a nonzero magnetic field, $h > 0$; the magnetization will then be positive.  As we decrease h to zero, the magnetization will discontinuously change from being positive for any $h>0$ to being negative for any $h < 0$.  So for $T<T_c$, there is a first-order phase transition line.  The phase diagram can be seen on this webpage and this is discussed more in most stat mech textbooks.  My favorite for this is Nigel Goldenfeld's Lectures On Phase Transitions And The Renormalization Group.
A: Let me give a more mathematical perspective on that.
As far as I know, the classical example of a system you are looking for -- is the basic landau theory with a cubic term:
$$F=r\Psi^2+s\Psi^4+\alpha\Psi^3,\quad r=r_0(T-T_c)$$


*

*If $\alpha=0$ then you have just a standard theory for 2nd order phase transistion.

*While at $\alpha\ne0$ you'll have a discontinuity in the order parameter dependence on temperature.
Regarding your comment: if one restricts values of $\Psi$ to be positive -- then one gets exactly what you want.  The restriction is quite natural if you, say, have two-component order parameter $\vec{\phi}=(\phi_1,\phi_2)$ and $\Psi=|\vec{\phi}|$.
A: Many order-disorder transitions are second order, for example the order-disorder transition in $\beta$-brass. So brass would be an example of a material showing both first and second order transitions.
However even simpler is, as Alexei pointed out, the magnetic transition in iron that happens at the Curie temperature is second order. So iron is probably the simplest system that fits your criteria.
