Generating high-quality responses is a key challenge for any open domain dialogue systems. However, even though there exist a variety of quality dimensions especially designed for dialogue evaluation (e.g., coherence and diversity scores), current dialogue systems rarely utilize them to guide the response generation during training. To alleviate this issue, we propose LSTDial (Long- and Short-Term Dialogue), a novel two-stage framework which generates and utilizes conversation evaluation as explicit feedback during training. Specifically, we fine-tune pre-trained dialogue systems through using turn-level quality feedback in the first stage and further train ever-improving dialogue agents through using dialogue-level quality feedback in the second stage. By using our approach on dialogue systems, capable of enabling dialogue generation with both short-term capabilities (generating more fluent, relevant and varied responses at the turn-level) and long-term capabilities (generating more coherent, engaging and informative responses at the dialogue-level). We implement LSTDial on four strong baseline models and experiment with two open-domain dialogue datasets. Experimental results show that LSTDial achieves significant improvement, enabling to generate better dialogue responses in terms of both human and automatic evaluation.