It sounds like a great start. When you aren't used to exercise just doing some is a huge help. You will want to increase intensity over time, but do it at a rate that is comfortable for your body. A common recommendation to avoid the repetitive stress injuries is to work up gradually, maybe 10% more a week.
I like the bike idea because it is joint friendly and it offers a mixed strength and endurance challenge. See if you can work up, over the next few months to 30 min of getting your heart rate up. Once you've done that, you might want to Google HIIT if you want to look at increasing intensity instead of time.
I am suspicious that you are doing tummy and arm exercises because you don't like these parts of your body. Sadly, you can't spot reduce. On the flip side, if you build a greater percentage of muscle to fat in your body as a whole you'll look thinner. A pound of muscle is smaller than a pound of fat. I would tend to think you would get better results doing compound exercises that work more muscle groups at once. That way you can develop more overall muscle on your chest, shoulders and back at the same time you shape your arms. So net net I would recommend push ups, rows, and shoulder press over arm isolation exercise. Feel free to do crunches too.
Do strength every other day, or say twice a week. The rest between sessions is critical, because that's when you actually rebuild the muscle. You can probably alternate days with the cardio if it's taking too much time to do it the same day. Start with what you can do pretty easily and do say 6 to 8 repetitions of it, then add a repetition each exercise session if you can, when you get to 12 then add another set. (12 push ups, rest 1 min, 8 more and so on) When you can do 2 sets of 12, make it harder (by adding weight or a tougher variation with bodyweight exercises like pushups)
bent over row (back and biceps):
both arms with dumbells or whatever weight you improvise http://www.youtube.com/watch?v=U-flna0-L-0
OR if you prefer, one armed at a timehttp://www.youtube.com/watch?v=EuTtX8k6UpE
notice they do NOT curve the back but keep their head up and look straight ahead...
OR a cheap option is surgical tubing or resistance bands when you are starting:http://www.youtube.com/watch?v=gKwZ6s71r5Y
lots of women neglect their back, but a strong back usually means better posture, which makes you look much better.push ups (chest, front shoulders, core stabilizers, triceps)progressions because I don't know what you can manage:
easy wall - http://www.youtube.com/watch?v=1RtM3L00BUU&feature=related
OR harder incline(stairs!)- http://www.youtube.com/watch?v=U3-QCCNpU6I
OR on the floor of course...
OR harder yet -http://www.youtube.com/watch?v=YS2agrYPFTgshoulder press (shoulders, triceps)
- Weights http://www.youtube.com/watch?v=8PQrNGqEFqg
OR Bands http://www.youtube.com/watch?v=fNl1oRWHUPw
...and disclaimer - I'm not a trainer, not a doctor etc. But this is pretty garden variety advice.