Your best bet is to buy an inexpensive Nespresso type machine. I've been looking for a good coffee machine and I bought a professional machine for nearly 900€. The other option would have been an Nespresso for about 100€. Buy one with a milk foamer. A little bit more expensive, but within your range. Why do I recommend this type of machine? Capsules! All your cups come out equal. No hassle. A little bit more expensive than grinding your own coffee.
The next best bet is consumer grade expresso machine from Krups (for instance), but then you have to grind your own coffee.
You need to know what coffee you want to drink. Your question is like asking "What do need to know before buying meat cooking tools". If you want to eat your meat as BBQ, you need a grill and tongs. If you want to eat it as meatloaf, you need a meat grinder and an oven. Similarly, with coffee, you need to know what coffee you want to drink. Different types of coffee are made with different processes, and each process needs different tools. You need a dripper and filters for drip coffee, an espresso machine for espresso, and so on.
Your decision will be probably based on
- taste and caffeine content. Coffee produced with different processes tastes differently. It also delivers different amounts of caffeine per ml of drink (although this can be tweaked within the same process). Espresso, drip coffee, Turkish coffee, mokka pot coffee, French press coffee, Aeropress coffee, instant coffee, Nespresso coffee and cold brewed coffee all produce a basic coffee drink (and this is not an exhaustive selection). If you are into mixed coffee drinks like mocha, latte machiato, cappucino, etc., you will need a combination of a tool which can produce the right coffee for your mixed drink and a tool which can create the drink (e.g. a way to foam milk).
- The money you want to invest. Many people love a good espresso, but a professional Italian machine costs several thousands of Euro, guzzles energy and takes up space. Home espresso automates offer a somewhat inferior result for a few hundreds of Euros. Or you can make Turkish coffee with a 2-Euro sieve and whatever small pot you already have in the kitchen. Most people go neither for the expensive espresso machines, nor for the Turkish method (which leaves dregs in the cup - most people in the West want their coffee clear) and choose some method which is affordable but still produces coffee of a type they find agreeable.
- The time you want to invest per cup. Nespresso has a devoted following of people for whom coffee making has to be convenient above all. There are methods which produce much better coffee for a fraction of the cost, but you have to 1) invest time in learning how to brew well, and 2) take the time to brew. It is mostly in the range of several minutes per cup, plus overhead for assembling and cleaning utensils.
If you are confused by the wide range of coffee methods and think "I don't know how each of these coffee types tastes like, how do I know which one to choose?" I would recommend to start by the second criterion (money) or third criterion (effort), depending on which one you are more pressed for. As an example, if you are a middle class person to whom it doesn't matter if they will spend 100 or 200 Euros on a tool which will be used as a part of their daily routine, go by effort. Look up what it takes to make coffee with the different methods, and think which you can imagine yourself doing. Then try coffee made with this method (if you need special tools for it, maybe you have a friend who has the tools?). If it is good enough for you, get the tools and start brewing. If not, move on to the next method on the list. Money-first would work similarly, but you rank the tools by price rather than effort per cup.
The first criterion - taste - is probably the one which is most likely to result in personal satisfaction, but you would have to know what you want before settling on a method, and it would be very difficult to find a way to taste all the different types of coffee and to develop a taste for each. Note that, if you are just starting out with coffee, your taste is also likely to develop and shift over time. So, I wouldn't recommend taste-first for people going for their first set of tools.
Best Answer
It sounds like the only thing you can do is use an interface disk - basically a piece of metal that is compatible with the induction cooktop, which will then transfer the heat to the espresso maker.
You can also potentially use an induction-compatible pan as an interface - see this question.
Unfortunately either way, you'll lose the fast-response benefit of the induction cooktop, because you'll now have to wait for the interface to heat.