Nowadays, most of the manufacturer are removing SD Card slot from smartphone due to decrease its thickness. To make people adapt this removal, smartphones are coming with big internal storage expansions which will solve your issue of low storage.
If you have a high budget then I will recomment you to buy Apple iPhone 6 or 6S with 128GB internal storage option. They both are the flagship smartphones of Apple and comes with impressive hardware specifications as well. iPhone 6S is an upgrade over iPhone 6 thats why it comes with better hardware specification and camera as well. While launching Apple iPhone 6S, they broke their tradition of 8-Mega Pixels of Camera and launched 6S with a bigger and better rear camera with 12-Mega Pixels. Apart from that it was launched with 2GB of RAM which was never seen in any Apple iPhone before that.
At the same Apple iPhone 6 comes with 1GB RAM and 8-Mega Pixels of camera. It is not good as Apple iPhone 6S but better than other smartphones with same camera specificatons. You also have 128GB storage option in iPhone 6.
High Budget - Android
If iOS is not your taste or you hate their restriction then you have another option of Samsung Galaxy S6.
Samsung Galaxy S6 is the flagship smartphone of Samsung and an addition in the S series as well. Samsung's S series is the most rated and high specifications series of Samsung as well. Samsung Galaxy S6 comes with very powerful hardware specifications and like iPhone 6's it comes with internal storage of 128GB (optional) at a premium. Apart from hardware Galaxy S6 comes with some other impressive features as well such as Nature resistant body (Water-and-Dust Proof), best-in-class display, etc.
If you are low on budget then you don't have many options. You can go for ASUS Zenfone series. Asus recently launched new smartphone - Asus Zenfone 2. It comes with storage option of 128GB. Looking at the price you are paid back about 2 times. You get a powerful Intel Atom Z3580 processor with 4GB RAM. Apart from that this smartphone features a 13-Mega Pixels of Rear Camera whereas you get a 5-Mega Pixels of front camera. Only disadvantage of this smartphone is its UI. It runs on Android OS but comes with cosmetic changes and bloatware which directly affects the performance of this smartphone. (You can solve these issues by rooting your smartphoine and installing a custom OS. After these modifications this is the best choice for you.)
If none of above smartphone suits your needs then you can wait for few months. Many more smartphones are expected to comes with high internal storage.
I am currently pursuing High School and planning to pursue career in Computer. Currently, I work on one of my blog - JTech Articles.