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LingakuLingaku続ける after the ます-stem of a verb means "keep doing": the action goes on without stopping, or someone sticks at it. Take the ます-form and swap ます for 続ける: 走ります → 走り続ける, 食べます → 食べ続ける, します → し続ける.
The dictionary form and the て-form don't attach: ×走る続ける, ×走って続ける.
The result is an ordinary る-verb, so it conjugates like one: 読み続けます, 読み続けた, 読み続けたい.
続ける is transitive: someone keeps something going. Its intransitive partner つづく続く is for things that go on by themselves, and 〜続く is common with weather and other things nobody is doing on purpose.
On its own, 続ける takes を and 続く takes が.
〜ている says an action is happening now. 続ける puts the weight on how long it has gone on without stopping.