We use the Galaxy Evolution Explorer (GALEX) Medium and All-Sky Imaging Survey (MIS and AIS) data from the first public data release (GR1), matched to the Sloan Digital Sky Survey (SDSS) DR3 catalog, to perform source classification. The GALEX surveys provide photometry in far- and near-UV bands and the SDSS in five optical bands (u, g, r, i, z). The GR1/DR3 overlapping areas are 363 (86) deg2 for the GALEX AIS (MIS), for sources within the 0.5° central area of the GALEX fields. Our sample covers mostly |b| > 30° Galactic latitudes. We present statistical properties of the GALEX-SDSS matched sources catalog, containing >2 × 106 objects detected in at least one UV band. We classify the matched sources by comparing the seven-band photometry to model colors constructed for different classes of astrophysical objects. For sources with photometric errors <0.3 mag, the corresponding typical AB-magnitude limits are mFUV ~ 21.5, mNUV ~ 22.5 for AIS, and mFUV ~ 24, mNUV ~ 24.5 for MIS. At AIS depth, the number of Galactic and extragalactic objects are comparable, but the latter predominate in the MIS. On the basis of our stellar models, we estimate the GALEX surveys detect hot white dwarfs throughout the Milky Way halo (down to a radius of 0.04 R☉ at MIS depth), providing an unprecedented improvement in the Galactic WD census. Their observed surface density is consistent with Milky Way model predictions. We also select low-redshift QSO candidates, extending the known QSO samples to lower magnitudes, and providing z ≈ 1 candidates for detailed z ≈ 1 follow-up investigations. SDSS optical spectra available for a large subsample confirm the classification for the photometrically selected candidates with 97% purity for single hot stars, 45% (AIS) or 31% (MIS) for binaries containing a hot star and a cooler companion, and about 85% for QSOs.