The "Space Age" will also be the "Rocket Age" since rockets will be the main means of traveling in space. There are various different kinds of rocket engine, but the traditional chemical rocket engine (in which a fuel and an oxidizer are mixed and the hot gases produced are ejected in the opposite direction to the direction of travel) will be the "work-horse" of life in space, like motor-car engines on Earth. To date most rocket engines are made to operate only once on an "expendable" vehicle, like a missile. So rocket operations are extremely expensive and small scale, and rocket engineering is still a small, specialized field.
In the Rocket Age, rocket engines and rocket vehicles will be "reusable" like all other forms of transportation. Railways, shipping lines, airlines etc don't use the word "reusable" because it's obvious. We don't talk about "reusable cars", "reusable buses", or "reusable aeroplanes". So it should and will be with space vehicles. So, far from being a small technical back-water, rocket engineering is still in its early days and has yet to see its heyday.
So we're looking forward to the era in which chemical rocket engines of many sizes and shapes are in everyday use - the "work-horses" of life in orbit. Without trying to predict the details such as what size and types of vehicles, what orbits, what types of engine and so on will be used, it is possible to foresee many of the broad outlines of the "rocket age". For example, rocket vehicles for launch, for in-orbit operations, and for longer trips such as to the Moon and back, will differ in many ways - though some may be multi-purpose, probably particularly at the early stages before the different markets are large enough to support more specialized vehicles. Specifically, vehicles used for launching from Earth to orbit and returning to Earth need to have aerodynamic and thermal protection which isn't needed for operations in space. Vehicles that operate only in space will have weird and wonderful shapes without concern for aerodynamics, but to have minimum mass among other constraints.
Another example is that although using LH2 (liquid hydrogen) / LOX (liquid oxygen) engines is in some respects more tricky than, say kerosene / LOX engines, it has the advantage that LH2 and LOX can be made by splitting water (H2O) using electricity. In orbit electricity can be generated from sunlight - so without using any fuel - and so wherever there's water you can make rocket fuel. This will be very useful in orbital operations, and we can envisage propellant stores comprising large insulated tanks for both LH2 and LOX, with insulated piping and pumps for recompressing gas to liquid, large water tanks, large solar panels for generating electricity, docks for receiving bulk supplies, exhangeable tanks, specialised delivery vehicles, pumps for supplying customer vehicles - whether passenger craft returning to Earth, small local orbit "taxis", cargo craft heading up to the Moon or further out, and others. The detailed design of these facilities to make them economical and competitive, raises loads of fascinating design issues. But with an orbital tourism business generating tens of $billions of turnover, it's easy to see that such business opportunities will exist. And from a business point of view they're far more promising, near-term and potentially profitable than plans for a possible government-financed base on Mars possibly 40 (?) years from now.