The "Calibre 38 Rimless Smokeless" ammunition this gun uses is called 38 ACP nowadays. (The ACP stands for Automatic Colt Pistol.) It is still made, or was until fairly recently, but only a well-stocked gun store will have it. You might be able to find it on the Internet, but shipping may be pricey - live ammo cannot be sent by US Mail.
Be careful - in the 1920's, Colt had a brainstorm and introduced the "38 Super" cartridge for the Model 1911 automatic pistol. It is dimensionally identical to 38 ACP - but loaded to a much higher pressure. Many guns like yours have been wrecked with it.
Whether it is a good idea to shoot your gun at all I will leave to people who know more about it. It is a Browning design, but it is very old, and was the first pistol he ever designed, or at least the first to be manufactured. He learned a lot as time went on.
PS - I know nothing about values, and cannot help you there. Your gun is a "1903 Pocket Hammer Model", and I believe it is the most common of Colt's 38 ACP "parallel ruler" guns.
At auction your gun would sell pretty quick for $500 and over time maybe $700. The above info is pretty much on target. The 1903 Hammer is a descendant of the Model 1900 and the 1902. It would be a great stroke of luck to find ammo in a gun store. The only ammo I have been able to get the past several years has been held by individuals. Have it checked by a knowledgable person before you shoot it. As above, it will shoot .38 Super but .38 Super is far too high pressure for the gun and may damage it.
Another area of possible damage is trying to drive out the key (Colt called it the slide stop, a term that confuses M1911 fans to whom it means something different). It is removed by simply pushing in on the concave plug below the muzzle and turning the gun on its left side, when the key will usually fall out.
BUT be VERY CAREFUL to put the key back in before firing; if it is left out, the slide will come right back off the frame and into the shooter's face!
Your gun was made in 1907. The US issue holster does not belong with the gun as none of the 1903 pocket hammer guns were ever adopted or used by any military service. In the condition shown, the gun would probably bring around $700-900.