Echoing some of the other posts, I think you will find that if you have the space and the installation wherewithal that a wire antenna - a standard (half wave) dipole will be your best first antenna. It shouldn't cost much for the materials and if you can get it installed high enough you should get very good performance.
As for bands, if you want lots of opportunity for DX and not take up too much space lengthwise (horizontally in your yard) and not have to go too high vertically, you should probably consider 20 Meters and if you can, also 40 Meters. With 20 Meters and 40 Meters you can probably operate around the clock (day and night) if the bands cooperate. With 20 and 40 you should be able to have plenty of opportunities to DX for SSB, data, and CW if you happen to do/enjoy those. If you don't have the horizontal space or you can't get the antennas high enough then there are other options including verticals but if you can do a wire antenna I'd go a dipole at least as a starter solution.
As for height, if you can get it a quarter-wave length up, that would be good and a half-wave length would be even better. I think higher will help your take off angle which will help you with DX, but there is another rule of thumb that says "a dipole should always be at least 20 feet higher than the height you were able to achieve."
http://www.csgnetwork.com/freqwavelengthcalc.htmlhttps://www.qsl.net/aa3rl/ant2.html20 Meters (exact wavelength depends on specific frequency within band)
Half wave length ~34 feet (horizontal space you will need for wire)
Quarter wave length ~17 feet (preferred minimum height above ground)
40 Meters (exact wavelength depends on specific frequency within band)
Half wave length ~68 feet (horizontal space you will need for wire)
Quarter wave length ~34 feet (preferred minimum height above ground)