Ideally you'd want a mix of cardio and strength work. HIIT for cardio (because its the most efficient). Free weights are good, compound lifts are best. There's already been a good discussion of the pros and cons of weights here so I won't try to get into that. Crossfit would be way too intense for you right now, but look up crossfit metcon workouts and see if you could manage to do stuff like that. You can design killer workouts by just doing a circuit of body weight workouts, great for cardio and muscle endurance. Some people already mentioned examples of workouts like this. Body weight circuits are great, and you don't need any equipment to do them either.
Don't know where your fitness is at right now, but about a year ago I could barely do a few knee push ups. None on my toes. So what I started doing was every time I got tired of sitting at my desk (or just thought I needed a break or something) I'd drop down and do a set of knee push ups. Started off with sets of 10, worked up to sets of 35-40 in about a month, I did basically as much as I felt I could do each time, trying to push further each time. Some days I'd manage to get to ~250ish push ups in total, others just around ~80, that's okay as long as it's consistent. Within a month I could do about 10 full push ups with good form. Might be a good way to approach to start. I think if you look online there are loads of "30 day squat challenge" and stuff like that.
In the end though, if you're interested in getting fitter (and being fit doesn't necessarily have anything to do with weight), you're going to want to find something you like so you can stick to it. There's no point in doing something you hate, because you're going to wait for the day you have to stop doing it. I've seen (overheard) loads of people at the gym talk about working out until they reach their target weight and hoping to stop afterwards, and it's those people that I haven't seen change shape at all. Food too though, if you're interested in weight (fat) loss then you gotta get your nutrition in check. It's about 80% diet and 20% exercise.
