Yes, you should be able to do either. The initial notice of appeal filing--I'm presuming you're in federal court from your question--is a simple one page form, which you should be able to find easily on the web. Absent a serious professional responsibility issue preventing your current lawyer from filing it, it seems at least a little odd that he won't file it. Even if you and he are parting ways for financial reasons, your unhappy with his work and even-or especially-if he thinks you have reason to be unhappy with his work at the trial level, it's best for him (and likely easiest for you) if he files it). But time permitting, if as it seems, you were in satisfied with his work and there's been a break down of the relationship, it seems just as well to have the lawyer who will be handling the appeal file the notice. You can do it pro se, but given the time frame you referenced, I suspect the trial court gave you 48 hours to file a notice of appeal and to move for a stay of at least some aspect of the decision below? Obtaining a stay pending an appeal is far more complicated than simply filing a notice of appeal. So if you are seeking a stay (or postponement of enforcement of an order) pending the outcome of the appeal, do consider retaining a qualified lawyer.
Unless the appeal is frivolous or there are certain unusual circumstances, unless the lawyer was retained explicitly to do the trial work only, it may be that he's technically required to file the notice if you direct him to (and you a pay the court filing fee) as a matter of professional responsibility, but I never think it's a good idea to have a lawyer feel forced to work on a case, nor does it seem you would want to use that lawyer under the circumstances. Deal with filing the notice, seeking a stay pending the determination of the appeal of that's what your looking for--and if you can hire counsel to deal with that as well as perfecting, briefing and arguing the appeal. If you want, and you believe your trial lawyer's conduct harmed you or was truly egregious you can contact the bar to make a formal complaint and/or consult with a malpractice lawyer to see if you've a claim to assert. But do note that the burden on the plaintiff in a malpractice case is a high one. In a nutshell you have to show that but for a specific reckless or negligent act, you would have unquestionably won your case.
Best of luck with your appeal, and if I can be of any assistance, please contact me.
Normally the trial attorney is suppossed to do it but I'm sure anyone lawyer or the defendant pro se can file and serve it.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.