You shouldn't have a problem taking photos of the moon with your phone, though lining up the camera with the eyepiece may be a fiddle.
It's likely that your camera will try and adjust the exposure as you're aiming and focusing, which means you've got a moving target in many ways (so to speak)
What makes life easier is that you can buy a cradle for your phone that clamps onto the eyepiece. This makes things easier as it stabilises what you're looking at, allows your camera time to get good focus and exposure.
Photographing planets is likely to be more problematic though, because to get them big enough to make out details on the image, you'll need high magnification, and this means a dimmer image. I took a photo of the recent Jupiter/Mars conjunction using a 102mm telescope (with 3x the effective magnification) and neither planet's image is particularly impressive. post134732.html#p134732
It can be a good idea to upgrade your eyepieces; you can get Plossl eyepieces at a reasonable cost, and which one to do first will depend on whether you enjoy looking at wide star fields using the 20mm EP, or smaller objects with the 10mm one. Do which ever one you appreciate the view from more.
If the 10mm EP, a barlow can be useful - your telescope has a focal ratio of f/4 (which means its focal length is 4x the aperture), and a useful rule of thumb is that you can usually use an eyepiece in mm down to the focal ratio. This means that a 2x barlow and a 9mm or 10mm should be usable (and in my experience, easier to use than an actual 5mm or 4mm eyepiece).
Celestron do a 2x Barlow with a t-thread adapter for about £25 - the t-thread will allow you to attach a dSLR in the future, though that's more likely to be with your next telescope perhaps.
If you have a local astronomy society, you could see if they do observing evenings. If you take your scope along you may be able to try other people's affordable eyepieces to see which ones you like the look of.