Digital cameras use sensors that actually have a fixed, intrinsic sensitivity to light. This means that as the total amount of incident light drops, so too does true exposure. Digital cameras get around this by adding configurable ISO levels, which simply boost the electronic signal generated by that incoming light by a certain amount. The only way to truly increase the strength of the signal is to let in more light...and the only way to let in more light is either with a longer exposure, or a larger aperture. Exposure can only go so far without tracking before stars start to "trail", so increasing the size of the aperture is the best option...if you have it.
To maximize the signal level of your image, you need to use the highest ISO you can get away with. If ISO 1600 isn't producing decent results, then increase it to ISO 3200, or even higher if your camera supports it natively, and a higher level doesn't introduce too much read noise.
For the most part, once a modern digital camera gets up to higher ISO levels, noise is dominated by photon shot noise, and read noise is minimal (less than 2-3e-). Regardless of which ISO setting you use, you will have to do further processing in post to maximize the results. You can either use some prodigious noise reduction in post, or you can try to take multiple frames and stack them using some kind of median averaging, to reduce the noise you see at high ISO.
To demonstrate, here is a shot of the summer milky way. The first is the original shot, without any processing, where as the second has been processed. You'll notice the difference in contrast, color, saturation. etc.
While it often doesn't appear so to our eyes, which have an amazing natural ability to adapt to changing ambient light levels (unlike a camera), stars are EXTREMELY dim. They emit the lowest levels of natural light, such that it has long been an intense area of research to develop new technology that is more sensitive to low light levels (i.e. black silicon, which has negligible read noise and has high sensitivity to the extremely dim light of stars.)
Faster lenses can be a huge bonus. You should try to find the sharpest f/2.8 or faster lenses possible for your astrophotography. Samyang lenses, which also sell under the name Rokinon and a number of others, have excellent sharpness from corner to corner, and have apertures as fast as f/1.4 for some of the more popular wide angles (14mm and 24mm, for example.) If you take your output size into account, you may be able to expose for longer as well. The images above were exposed for 30 sec. @ f/2.8 ISO 1600, and at full size, they exhibit a small amount of star trailing. Stars look like natural points of light scaled down for presentation here on the web, and I could have probably gotten away with 45-50 seconds or more worth of exposure time. If you only ever share your photos online at 1024 pixels wide, you might try exposing for longer, and not worry about the slight startrailing that occurs.