Ionian is the same as the major scale or the same as playing all the white keys from C to C. All major scales are Ionian scales, just starting on different notes than C.
Locrian is the same as a minor scale with a lowered 2nd step and a lowered 5th step or the white keys from B to B. To start on a different note, play the minor scale and lower the 2nd and 5th steps. For example, B minor has two sharps, F and C. Since C is the 2nd step and F is the 5th, both are lowered to natural.
I'm breaking my pattern and discussing two modes at once (gasp!) in this post. The reason is simple; I don't have a lot to say about either of them.
Ionian is major. Major is Ionian. There just isn't much else to say on the subject. It is possible to argue that since in modes, you don't modulate much (except from mode to mode) that the lack of shifting key signatures is what makes a piece modal Ionian rather than major. But this is a weak argument since there are plenty of major pieces that don't change keys much. It is also possible to argue that how Ionian is used with the OTHER modes makes it Ionian but this doesn't really change how melodies and harmonies in Ionian itself work.
Now on to Locrian. Locrian is the one mode I don't get. At all. Not even a little. It starts and stops on the 7th step of the major scale which is one of the most unresolved sounds possible. In addition, moving both the 2nd and the 5th takes away any sense of center this mode might have had. To me, it always sounds wrong, unfinished, like the composer is messing with the audience or possibly trying (and failing) to impress a theory teacher. Speaking of which, I did have a theory teacher who claimed Locrian was her favorite mode and played it at the end of class regularly. It almost always caused us to run to the nearest piano and play the note after the last note of the Locrian mode just to get some sense of resolution. We may have been too steeped in the Western scales to adapt, I don't know. I have considered adjusting this scale to see if that made it more palatable to me (and I may very well do this) but the simple truth is that it won't be Locrian any more. It would become a scale outside of the modal system. Of course, there are many scales that don't fit into the modal system and they are well worth exploring. Generally, just like the modes, they all work differently and different people like different ones.
So if you enjoy that empty, tension filled feeling of a musical line hanging in mid-air, run with it! It could possibly fit in with some Jazz styles which end on non-tonic chords. But you will have to figure out how to work this mode without help from me!